Within Tibetan Buddhism, are the Lamas always the same "soul"?
Because Buddhism denies the notion of "self", it does not have a concept of Avatar, as someone literally being a reincarnation of the same "self".

Tibetan Buddhism does have a notion of Tulku though, which is similar. Tulku is a new person that inherits the life stories and experiences of a previous person. This is done through teaching this person to deliberately identify with their predecessor, to learn predecessor's history, and to carry on his or her work.

Because the new person's life is so heavily influenced and inspired by the predecessor, we can say that the new person is an embodiment of the abstract energy of the previous one.

The case with your specific question, any person, and even multiple people, regardless of status or title or role, could be designated as Tulkus of Dalai Lama, but to truly qualify as such, the person would need to seriously work on their identity as an embodiment of Dalai Lama's heritage.