How to change Auto Layout constraints after they are set when using constraintEqualToAnchor()?

You need to deactivate the previous constraint when activating a new one so that you don't end up over constraining your view. To do that, store a reference to each of the constraints as a property in your ViewController and then set the active property of the old constraint to false before creating and activating the new constraint:

Swift 2.x:

class ViewController: UIViewController {
    var leftConstraint: NSLayoutConstraint?
    var trailingConstraint: NSLayoutConstraint?
    var topConstraint: NSLayoutConstraint?
    var bottomConstraint: NSLayoutConstraint?

    override func viewDidLoad() {
        super.viewDidLoad()

        let myView = UIView()
        myView.backgroundColor = UIColor.orangeColor()
        myView.translatesAutoresizingMaskIntoConstraints = false
        view.addSubview(myView)

        leftConstraint = myView.leftAnchor.constraintEqualToAnchor(view.leftAnchor)
        leftConstraint?.active = true

        trailingConstraint = myView.trailingAnchor.constraintEqualToAnchor(view.trailingAnchor)
        trailingConstraint?.active = true

        topConstraint = myView.topAnchor.constraintEqualToAnchor(view.topAnchor)
        topConstraint?.active = true

        bottomConstraint = myView.bottomAnchor.constraintEqualToAnchor(view.bottomAnchor)
        bottomConstraint?.active = true

        /******************************************/
        /* I try to change one of the constraints */
        /******************************************/
        leftConstraint?.active = false
        leftConstraint = myView.leftAnchor.constraintEqualToAnchor(view.rightAnchor, constant: -100)
        leftConstraint?.active = true
    }
}

---

Update for Swift 3 syntax:

class ViewController: UIViewController {
    var leftConstraint: NSLayoutConstraint?
    var trailingConstraint: NSLayoutConstraint?
    var topConstraint: NSLayoutConstraint?
    var bottomConstraint: NSLayoutConstraint?

    override func viewDidLoad() {
        super.viewDidLoad()

        let myView = UIView()
        myView.backgroundColor = UIColor.orange
        myView.translatesAutoresizingMaskIntoConstraints = false
        view.addSubview(myView)

        leftConstraint = myView.leftAnchor.constraint(equalTo: view.leftAnchor)
        leftConstraint?.isActive = true

        trailingConstraint = myView.trailingAnchor.constraint(equalTo: view.trailingAnchor)
        trailingConstraint?.isActive = true

        topConstraint = myView.topAnchor.constraint(equalTo: view.topAnchor)
        topConstraint?.isActive = true

        bottomConstraint = myView.bottomAnchor.constraint(equalTo: view.bottomAnchor)
        bottomConstraint?.isActive = true

        /******************************************/
        /* I try to change one of the constraints */
        /******************************************/
        leftConstraint?.isActive = false
        leftConstraint = myView.leftAnchor.constraint(equalTo: view.rightAnchor, constant: -100)
        leftConstraint?.isActive = true
    }
}

Here is an example declaring a constraint c which will be referenced later in time. We set a new constant value and then call layout on the superview.

myView.translatesAutoresizingMaskIntoConstraints = false

var constraints: [NSLayoutConstraint] = [
    myView.topAnchor.constraintEqualToAnchor(view.topAnchor),
    myView.leftAnchor.constraintEqualToAnchor(view.leftAnchor),
    myView.bottomAnchor.constraintEqualToAnchor(view.bottomAnchor)
]         
let c = myView.rightAnchor.constraintEqualToAnchor(view.rightAnchor)       
constraints.append(c)   

view.addSubview(myView)
NSLayoutConstraint.activateConstraints(constraints)

// Some time later
c.constant = -100
view.setNeedsLayout()