Two team members have recently approached my manager (or in truth my manager’s manager) with the same predicament. They went about in different ways, the first one tactfully explaining what was missing for them in their current role, the second one just more of a rant.
The outcome was pretty much the same, both got their wishes and were allowed to move on without any apparent prejudice. Although without doubt the first won the professional respect stakes.
But, all in all I think it depends very much in the organisational culture and your direct management team. As a manager i would be more helpful to the individual who appears calm and has carefully thought through what is missing and what they are looking far in their next move.