Is there a public API for card view UI that can be seen across iOS 10?

Question

You can build the segue in interface builder. Selecting modal segue from ViewController to CardViewController.

For your CardViewController :

import UIKit

class CardViewController: UIViewController {

  required init?(coder aDecoder: NSCoder) {
    super.init(coder: aDecoder)
    self.commonInit()
  }

  override init(nibName nibNameOrNil: String!, bundle nibBundleOrNil: Bundle!)  {
    super.init(nibName: nibNameOrNil, bundle: nibBundleOrNil)

    self.commonInit()
  }

  func commonInit() {
    self.modalPresentationStyle = .custom
    self.transitioningDelegate = self
  }

  override func viewDidLoad() {
    super.viewDidLoad()

    roundViews()
  }

  func roundViews() {
    view.layer.cornerRadius = 8
    view.clipsToBounds = true
  }

}

then add this extension:

extension CardViewController: UIViewControllerTransitioningDelegate {

  func presentationController(forPresented presented: UIViewController, presenting: UIViewController?, source: UIViewController) -> UIPresentationController? {
    if presented == self {
      return CardPresentationController(presentedViewController: presented, presenting: presenting)
    }
    return nil
  }

  func animationController(forPresented presented: UIViewController, presenting: UIViewController, source: UIViewController) -> UIViewControllerAnimatedTransitioning? {
    if presented == self {
      return CardAnimationController(isPresenting: true)
    } else {
      return nil
    }
  }

  func animationController(forDismissed dismissed: UIViewController) -> UIViewControllerAnimatedTransitioning? {
    if dismissed == self {
      return CardAnimationController(isPresenting: false)
    } else {
      return nil
    }
  }

}

Finally, you will need 2 more classes:

import UIKit

class CardPresentationController: UIPresentationController {

  lazy var dimmingView :UIView = {
    let view = UIView(frame: self.containerView!.bounds)
    view.backgroundColor = UIColor(red: 0.0, green: 0.0, blue: 0.0, alpha: 0.3)
    view.layer.cornerRadius = 8
    view.clipsToBounds = true
    return view
  }()

  override func presentationTransitionWillBegin() {

    guard
      let containerView = containerView,
      let presentedView = presentedView
      else {
        return
    }

    // Add the dimming view and the presented view to the heirarchy
    dimmingView.frame = containerView.bounds
    containerView.addSubview(dimmingView)
    containerView.addSubview(presentedView)

    // Fade in the dimming view alongside the transition
    if let transitionCoordinator = self.presentingViewController.transitionCoordinator {
      transitionCoordinator.animate(alongsideTransition: {(context: UIViewControllerTransitionCoordinatorContext!) -> Void in
        self.dimmingView.alpha = 1.0
      }, completion:nil)
    }
  }

  override func presentationTransitionDidEnd(_ completed: Bool)  {
    // If the presentation didn't complete, remove the dimming view
    if !completed {
      self.dimmingView.removeFromSuperview()
    }
  }

  override func dismissalTransitionWillBegin()  {
    // Fade out the dimming view alongside the transition
    if let transitionCoordinator = self.presentingViewController.transitionCoordinator {
      transitionCoordinator.animate(alongsideTransition: {(context: UIViewControllerTransitionCoordinatorContext!) -> Void in
        self.dimmingView.alpha  = 0.0
      }, completion:nil)
    }
  }

  override func dismissalTransitionDidEnd(_ completed: Bool) {
    // If the dismissal completed, remove the dimming view
    if completed {
      self.dimmingView.removeFromSuperview()
    }
  }

  override var frameOfPresentedViewInContainerView : CGRect {

    // We don't want the presented view to fill the whole container view, so inset it's frame
    let frame = self.containerView!.bounds;
    var presentedViewFrame = CGRect.zero
    presentedViewFrame.size = CGSize(width: frame.size.width, height: frame.size.height - 40)
    presentedViewFrame.origin = CGPoint(x: 0, y: 40)

    return presentedViewFrame
  }

  override func viewWillTransition(to size: CGSize, with transitionCoordinator: UIViewControllerTransitionCoordinator) {
    super.viewWillTransition(to: size, with: transitionCoordinator)

    guard
      let containerView = containerView
      else {
        return
    }

    transitionCoordinator.animate(alongsideTransition: {(context: UIViewControllerTransitionCoordinatorContext!) -> Void in
      self.dimmingView.frame = containerView.bounds
    }, completion:nil)
  }

}

and:

import UIKit


class CardAnimationController: NSObject {

  let isPresenting :Bool
  let duration :TimeInterval = 0.5

  init(isPresenting: Bool) {
    self.isPresenting = isPresenting

    super.init()
  }
}

// MARK: - UIViewControllerAnimatedTransitioning

extension CardAnimationController: UIViewControllerAnimatedTransitioning {

  func transitionDuration(using transitionContext: UIViewControllerContextTransitioning?) -> TimeInterval {
    return self.duration
  }

  func animateTransition(using transitionContext: UIViewControllerContextTransitioning)  {
    let fromVC = transitionContext.viewController(forKey: UITransitionContextViewControllerKey.from)
    let fromView = fromVC?.view
    let toVC = transitionContext.viewController(forKey: UITransitionContextViewControllerKey.to)
    let toView = toVC?.view

    let containerView = transitionContext.containerView

    if isPresenting {
      containerView.addSubview(toView!)
    }

    let bottomVC = isPresenting ? fromVC : toVC
    let bottomPresentingView = bottomVC?.view

    let topVC = isPresenting ? toVC : fromVC
    let topPresentedView = topVC?.view
    var topPresentedFrame = transitionContext.finalFrame(for: topVC!)
    let topDismissedFrame = topPresentedFrame
    topPresentedFrame.origin.y -= topDismissedFrame.size.height
    let topInitialFrame = topDismissedFrame
    let topFinalFrame = isPresenting ? topPresentedFrame : topDismissedFrame
    topPresentedView?.frame = topInitialFrame

    UIView.animate(withDuration: self.transitionDuration(using: transitionContext),
                   delay: 0,
                   usingSpringWithDamping: 300.0,
                   initialSpringVelocity: 5.0,
                   options: [.allowUserInteraction, .beginFromCurrentState], //[.Alert, .Badge]
      animations: {
        topPresentedView?.frame = topFinalFrame
        let scalingFactor : CGFloat = self.isPresenting ? 0.92 : 1.0
        bottomPresentingView?.transform = CGAffineTransform.identity.scaledBy(x: scalingFactor, y: scalingFactor)

    }, completion: {
      (value: Bool) in
      if !self.isPresenting {
        fromView?.removeFromSuperview()
      }
    })


    if isPresenting {
      animatePresentationWithTransitionContext(transitionContext)
    } else {
      animateDismissalWithTransitionContext(transitionContext)
    }
  }

  func animatePresentationWithTransitionContext(_ transitionContext: UIViewControllerContextTransitioning) {

    let containerView = transitionContext.containerView
    guard
      let presentedController = transitionContext.viewController(forKey: UITransitionContextViewControllerKey.to),
      let presentedControllerView = transitionContext.view(forKey: UITransitionContextViewKey.to)
      else {
        return
    }

    // Position the presented view off the top of the container view
    presentedControllerView.frame = transitionContext.finalFrame(for: presentedController)
    presentedControllerView.center.y += containerView.bounds.size.height

    containerView.addSubview(presentedControllerView)

    // Animate the presented view to it's final position
    UIView.animate(withDuration: self.duration, delay: 0.0, usingSpringWithDamping: 1.0, initialSpringVelocity: 0.0, options: .allowUserInteraction, animations: {
      presentedControllerView.center.y -= containerView.bounds.size.height
    }, completion: {(completed: Bool) -> Void in
      transitionContext.completeTransition(completed)
    })
  }

  func animateDismissalWithTransitionContext(_ transitionContext: UIViewControllerContextTransitioning) {

    let containerView = transitionContext.containerView
    guard
      let presentedControllerView = transitionContext.view(forKey: UITransitionContextViewKey.from)
      else {
        return
    }

    // Animate the presented view off the bottom of the view
    UIView.animate(withDuration: self.duration, delay: 0.0, usingSpringWithDamping: 1.0, initialSpringVelocity: 0.0, options: .allowUserInteraction, animations: {
      presentedControllerView.center.y += containerView.bounds.size.height
    }, completion: {(completed: Bool) -> Void in
      transitionContext.completeTransition(completed)
    })
  }
}

Finally, in order to animate the CardViewController closing, hook your closing button to FirstResponder selecting dismiss and add this method to ViewController:

func dismiss(_ segue: UIStoryboardSegue) {
    self.dismiss(animated: true, completion: nil)
}

Apple show how to do this using UIViewPropertyAnimator in WWDC 2017 Session 230: Advanced Animations with UIKit

The basic idea is that you add a child view controller, and position it mostly off-screen. When tapped/panned you animate the child view controller's frame.

import UIKit

class CardViewController: UIViewController {
    override func viewDidLoad() {
        super.viewDidLoad()

        view.backgroundColor = .red
    }
}

class ViewController: UIViewController {
    private let cardViewController = CardViewController()
    private var cardHiddenConstraint: NSLayoutConstraint!
    private var cardVisibleConstraint: NSLayoutConstraint!

    override func viewDidLoad() {
        super.viewDidLoad()

        addChild(cardViewController)
        let cardViewControllerView = cardViewController.view!
        cardViewControllerView.translatesAutoresizingMaskIntoConstraints = false
        view.addSubview(cardViewControllerView)

        cardHiddenConstraint = cardViewControllerView.topAnchor.constraint(equalTo: view.bottomAnchor, constant: -50)
        cardVisibleConstraint = cardViewControllerView.topAnchor.constraint(equalTo: view.topAnchor, constant: 50)

        let cardViewControllerViewConstraints = [
            cardViewControllerView.leadingAnchor.constraint(equalTo: view.leadingAnchor),
            cardViewControllerView.trailingAnchor.constraint(equalTo: view.trailingAnchor),
            cardHiddenConstraint!,
            cardViewControllerView.heightAnchor.constraint(equalTo: view.heightAnchor)
        ]
        NSLayoutConstraint.activate(cardViewControllerViewConstraints)

        cardViewController.didMove(toParent: self)

        let tapGestureRecognizer = UITapGestureRecognizer(target: self, action: #selector(handleTapGesture(_:)))
        cardViewController.view.addGestureRecognizer(tapGestureRecognizer)
    }

    @objc private func handleTapGesture(_ gestureRecognizer: UITapGestureRecognizer) {
        let frameAnimator = UIViewPropertyAnimator(duration: 0.3, dampingRatio: 1) {
            if self.cardHiddenConstraint.isActive {
                self.cardHiddenConstraint.isActive = false
                self.cardVisibleConstraint.isActive = true
            } else {
                self.cardVisibleConstraint.isActive = false
                self.cardHiddenConstraint.isActive = true
            }
            self.view.layoutIfNeeded()
        }
        frameAnimator.startAnimation()
    }
}

Ok, I'll try to give you a compact solution with a minimum of code.

Fast solution. You need to present a controller modally with modalPresentationStyle-property set to .overCurrentContext. You can set the value before preset(controller:...)-method get called or in prepare(for:...)-one if it's a segue transition. For sliding up use modalTransitionStyle set to .coverVertical.

To "zoom out" source view just update its bounds in viewWill(Diss)appear-methods. In most of cases this will work.

Don't forget to set your modal controller background view transparent so the underlying view still be visible.

Sliding up/down smoothly. You need to setup a transition between the controllers in a proper way. If you look closer to Apple music app, you'll see a way to hide top controller with slide down gesture. You can customise your view (dis)appearance too. Take a look at this article. It uses UIKit-methods only. Unfortunately this way requires lots of code, but you can use 3rd party libraries to setup transitions. Like this one.