Swap rootViewController with animation?

You can use UIView.transition(with: view) to replace the rootViewController of a UIWindow:

guard let window = UIApplication.shared.keyWindow else {
    return
}

let storyboard = UIStoryboard(name: "Main", bundle: nil)
let vc = storyboard.instantiateViewController(withIdentifier: "MainTabbar")

// Set the new rootViewController of the window.
// Calling "UIView.transition" below will animate the swap.
window.rootViewController = vc

// A mask of options indicating how you want to perform the animations.
let options: UIView.AnimationOptions = .transitionCrossDissolve

// The duration of the transition animation, measured in seconds.
let duration: TimeInterval = 0.3

// Creates a transition animation.
// Though `animations` is optional, the documentation tells us that it must not be nil. ¯\_(ツ)_/¯
UIView.transition(with: window, duration: duration, options: options, animations: {}, completion:
{ completed in
    // maybe do something on completion here
})

An alternative solution:

let stb = UIStoryboard(name: "YOUR_STORYBOARD_NAME", bundle: nil)
let rootVC = stb.instantiateViewController(withIdentifier: "YOUR_TABBAR_VIEWCONTROLLER_NAME")
let snapshot = (UIApplication.shared.keyWindow?.snapshotView(afterScreenUpdates: true))!
rootVC.view.addSubview(snapshot)

UIApplication.shared.keyWindow?.rootViewController = rootVC
UIView.transition(with: snapshot, 
                  duration: 0.4,
                  options: .transitionCrossDissolve,
                  animations: { 
                      snapshot.layer.opacity = 0
                  },
                  completion: { status in 
                      snapshot.removeFromSuperview()
                  })

Swift 4

Paste function into AppDelegate:

func setRootViewController(_ vc: UIViewController, animated: Bool = true) {
    guard animated, let window = self.window else {
        self.window?.rootViewController = vc
        self.window?.makeKeyAndVisible()
        return
    }

    window.rootViewController = vc
    window.makeKeyAndVisible()
    UIView.transition(with: window,
                      duration: 0.3,
                      options: .transitionCrossDissolve,
                      animations: nil,
                      completion: nil)
}