Let's take a look at the grabbed and immobilized conditions.
Being grabbed means a creature is immobilized. Maintaining this condition on the creature occupies whatever appendage, object, or effect the grabber used to initiate the grab. This condition ends immediately on the creature if the grabber is subjected to an effect that prevents it from taking actions, or if the creature ends up outside the range of the grabbing power or effect.
When a creature is immobilized, it can’t move, unless it teleports or is pulled, pushed, or slid.
Being immobilized means you cannot move of your own will. So if an attack power allows you to move or shift, you are unable to do so. If the power says you can teleport, you are able to. And that will break the grab.
There is nothing special about the immobilized condition in this case, it is what it is regardless of it being from a grab or from some other source. The only special rule here is for escaping a grab, which allows the target to end the immobilized condition using a move action - Escape.
To reiterate, escaping is a move action, but that doesn't mean that it can be done as part of another move. It just means it takes up your move action out of the normal standard, move, minor you are allotted with each round.
Escape: Move Action
Acrobatics or Athletics: Make an Acrobatics check vs. Reflex or an Athletics check vs. Fortitude against
the creature or effect that immobilized you.
Success: You end the grab and can shift as part of this move action.
Failure: You’re still grabbed.
So once you've attempted an Escape, you've used your move action. If you succeeded, great, you can shift one square and are free to continue your turn. If not, you can try again by sacrificing your standard action for another move.
The only way to use a movement power to break a grab is if the grabbing creature is moved. If they are moved further away from their grabbed target than they can reach, the grab ends because they cannot maintain it.