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It really does work just the way I described, if you have set up the dependencies.

Create two tasks:

Use the Connection button to make a Finish->Start dependency between them:

Now drag the finish date of the prerequisite later, and watch the dependent task shift:

However, if you have not correctly established the dependency relationship between the two tasks (especially if you haven't assigned them to the same resource), you can drag the finish over the subsequent start without any complaint:

If you've assigned them to the same resource but haven't set up the dependency link, the dependent-task-that-isn't will be rescheduled to follow the new finish date when you next level the project.