Looks like it's pretty smooth on 360:
This time we have the talented bunch at Nixxes taking point on the Xbox 360 port and the results are remarkable. This is the same game that we enjoyed on Xbox One with nips and tucks made in all of the expected places. It lacks much of the visual panache of the next-gen version, but it's still an attractive title and a worthy sequel. It doesn't feel like an afterthought at all - it's a project that has received a lot of care and attention.
In fact, there is a strong argument that in one sense at least, the
Xbox 360 game is actually more playable than its current-gen sibling.
On Xbox One, we complained about input latency and its impact on the
game experience(...) Moving to Xbox 360, we were stunned to discover
that this same issue does not exist on Microsoft's older console to
anything like the same extent.
In fact, Rise of the Tomb Raider feels significantly more responsive
here on Xbox 360
Looks like you chose the correct game NOT to move into next-gen