iOS 7 screen edge gesture swipe from right to left (similar to edge swiping from left to right) on UINavigationController. *only* the edge swipe uses the custom transition; everything else uses default behaviors.

SlideAnimatedTransitioning.h

//
//  SlideAnimatedTransitioning.h
//  SwipeLeft
//
//  Created by Visnu on 4/14/14.
//  Copyright (c) 2014 Visnu Pitiyanuvath. All rights reserved.
//

#import <Foundation/Foundation.h>

@interface SlideAnimatedTransitioning : NSObject <UIViewControllerAnimatedTransitioning>

@end

SlideAnimatedTransitioning.m

//
//  SlideAnimatedTransitioning.m
//  SwipeLeft
//
//  Created by Visnu on 4/14/14.
//  Copyright (c) 2014 Visnu Pitiyanuvath. All rights reserved.
//

#import "SlideAnimatedTransitioning.h"

// Tries to duplicate the default iOS 7 slide transition for UINavigationController
@implementation SlideAnimatedTransitioning

- (void)animateTransition:(id<UIViewControllerContextTransitioning>)transitionContext
{
  UIView *containerView = [transitionContext containerView],
         *fromView = [transitionContext viewControllerForKey:UITransitionContextFromViewControllerKey].view,
         *toView   = [transitionContext viewControllerForKey:UITransitionContextToViewControllerKey].view;

  CGFloat width = containerView.frame.size.width;

  CGRect offsetLeft = fromView.frame; offsetLeft.origin.x = -width/3;

  CGRect offscreenRight = toView.frame; offscreenRight.origin.x = width;
  toView.frame = offscreenRight;
  toView.layer.shadowRadius = 5;
  toView.layer.shadowOpacity = 0.4;

  [containerView addSubview:toView];

  [UIView animateWithDuration:[self transitionDuration:transitionContext] delay:0 options:UIViewAnimationOptionCurveLinear animations:^{
    toView.frame = fromView.frame;

    fromView.frame = offsetLeft;
    fromView.layer.opacity = 0.9;
  } completion:^(BOOL finished) {
    fromView.layer.opacity = 1;
    toView.layer.shadowOpacity = 0;
    // when cancelling or completing the animation, ios simulator seems to sometimes flash black backgrounds during the animation. on devices, this doesn't seem to happen though.
    // containerView.backgroundColor = [UIColor whiteColor];
    [transitionContext completeTransition:![transitionContext transitionWasCancelled]];
  }];
}

- (NSTimeInterval)transitionDuration:(id<UIViewControllerContextTransitioning>)transitionContext
{
  return 0.2;
}

@end

ViewController.m

//
//  ViewController.m
//  SwipeLeft
//
//  Created by Visnu on 4/10/14.
//  Copyright (c) 2014 Visnu Pitiyanuvath. All rights reserved.
//

#import "ViewController.h"
#import "SlideAnimatedTransitioning.h"

@interface ViewController ()

@property UIScreenEdgePanGestureRecognizer *edgePanGestureRecognizer;
@property UIPercentDrivenInteractiveTransition *percentDrivenInteractiveTransition;

@end

@implementation ViewController

- (void)viewDidLoad
{
  [super viewDidLoad];

  // Setup a screen edge pan gesture recognizer to trigger from the right edge of the screen.
  self.edgePanGestureRecognizer = [[UIScreenEdgePanGestureRecognizer alloc] initWithTarget:self action:@selector(handleScreenEdgePanGesture:)];
  self.edgePanGestureRecognizer.edges = UIRectEdgeRight;
  [self.view addGestureRecognizer:self.edgePanGestureRecognizer];
}

- (void)handleScreenEdgePanGesture:(UIScreenEdgePanGestureRecognizer *)sender
{
  CGFloat width = self.view.frame.size.width,
          percent = MAX(-[sender translationInView:self.view].x, 0) / width;

  switch (sender.state) {
    case UIGestureRecognizerStateBegan:
      // Configure the navigation controller to return the custom (interactive/animated) transition.
      self.navigationController.delegate = self;
      // Trigger a segue (or push another view controller, whatevs you want).
      [self performSegueWithIdentifier:@"next" sender:sender];
      break;
    case UIGestureRecognizerStateChanged:
      // Update the transition using a UIPercentDrivenInteractiveTransition.
      [self.percentDrivenInteractiveTransition updateInteractiveTransition:percent];
      break;
    case UIGestureRecognizerStateEnded:
      // Cancel or finish depending on how the gesture ended.
      if (percent > 0.5 || fabs([sender velocityInView:self.view].x) > 1000)
        [self.percentDrivenInteractiveTransition finishInteractiveTransition];
      else
        [self.percentDrivenInteractiveTransition cancelInteractiveTransition];
      break;
    default:
      NSLog(@"unhandled state for gesture=%@", sender);
  }
}

- (id<UIViewControllerAnimatedTransitioning>)navigationController:(UINavigationController *)navigationController animationControllerForOperation:(UINavigationControllerOperation)operation fromViewController:(UIViewController *)fromVC toViewController:(UIViewController *)toVC
{
  // Have to return a custom transition here in order for the navigation controller to ask about the interactive stuff below. The SlideAnimatedTransitioning just tries to duplicate the default transition as much as possible.
  return [SlideAnimatedTransitioning new];
}

- (id<UIViewControllerInteractiveTransitioning>)navigationController:(UINavigationController *)navigationController interactionControllerForAnimationController:(id<UIViewControllerAnimatedTransitioning>)animationController
{
  // Unset the delegate so that all other types of transitions (back, normal animated push not initiated by a gesture) get the default behavior.
  self.navigationController.delegate = nil;

  if (self.edgePanGestureRecognizer.state == UIGestureRecognizerStateBegan) {
    self.percentDrivenInteractiveTransition = [UIPercentDrivenInteractiveTransition new];
    self.percentDrivenInteractiveTransition.completionCurve = UIViewAnimationOptionCurveEaseOut;
  } else {
    self.percentDrivenInteractiveTransition = nil;
  }

  return self.percentDrivenInteractiveTransition;
}

@end