Showing pushviewcontroller animation look like presentModalViewController

If you want to a fade animation, this approach works.

CATransition* transition = [CATransition animation];
transition.duration = 0.3;
transition.type = kCATransitionFade;
transition.subtype = kCATransitionFromTop;

[self.navigationController.view.layer addAnimation:transition forKey:kCATransition];
[self.navigationController pushViewController:gridController animated:NO];

Try this :

UIViewController *yourViewController = [[UIViewController alloc]init];

[UIView  beginAnimations: @"Showinfo"context: nil];
[UIView setAnimationCurve: UIViewAnimationCurveEaseInOut];
[UIView setAnimationDuration:0.75];
[self.navigationController pushViewController: yourViewController animated:NO];
[UIView setAnimationTransition:UIViewAnimationTransitionFlipFromRight forView:self.navigationController.view cache:NO];
[UIView commitAnimations];

For display PushViewConttroler animation as Present below code is working,

For Push

 ViewController *VC = [self.storyboard instantiateViewControllerWithIdentifier:@"ViewController"];
 CATransition* transition = [CATransition animation];
 transition.duration = 0.4f;
 transition.type = kCATransitionMoveIn;
 transition.subtype = kCATransitionFromTop;
 [self.navigationController.view.layer addAnimation:transition
                                                    forKey:kCATransition];
 [self.navigationController pushViewController:VC animated:NO];

And for POP

CATransition* transition = [CATransition animation];
transition.duration = 0.4f;
transition.type = kCATransitionReveal;
transition.subtype = kCATransitionFromBottom;
[self.navigationController.view.layer addAnimation:transition
                                            forKey:kCATransition];
[self.navigationController popViewControllerAnimated:NO];

There is 1 issue though, that its display light black background when its going up/down,
Working on that, As soon as its done post new code here.

For All Trying in Xamarin (C#) equivalent Code From Answer by @hardik Thakkar above, Has the issue of Black backgound during animation, rest all good.

For POP

    CATransition transition = CATransition.CreateAnimation();
    transition.Duration = 0.4;
    transition.Type = CAAnimation.TransitionReveal;
    transition.Subtype = CAAnimation.TransitionFromTop;
    this.NavigationController.View.Layer.AddAnimation(transition, nameof(CATransition));

For Push

    CATransition transition = CATransition.CreateAnimation();
    transition.Duration = 0.4;
    transition.Type = CAAnimation.TransitionReveal;
    transition.Subtype = CAAnimation.TransitionFromBottom;
    this.NavigationController.View.Layer.AddAnimation(transition, nameof(CATransition));