I think kamma take all rebirth ways at the same moment, litteral and logical, both are true.
Also we can not say that the next life would be "ME", because "ME" is the focus point of this very life experiance. This "ME" will die with this body. Future body will have different experiance, so he will have different conditioning, will have different personality. Different from me.
But it's true, only until i dont remember past-existances -experiance, when i remember this experiance, it become me, cause i'am awere of this experiance. Actualy remembering existances is just an another experiance, and like all experiance have conditioning effect on "me".
But until we dont remember, it can not be me. It's a kamma vawe with relay surfing
There is no nouns, there is verbs.
There is no "me", there is kamma.