@One Degree
There are cases where migration happened through prolonged periods.
To take an example 'm directly familiar with.
Persians and specifically my tribe the Hazzar tribe. We migrated from north western Ukraine east to Russia around 5000-7000 years ago(estimates by scholars)(and it happened in waves as far as told) , then south towards the Iranian plateau around 4500 years ago.
If we were to take skin color as the prime example of the adoption process, we had much lighter skin color than the Roujams who were in the southern half of the Iranian plateau and had far older presence than we did there, and we can know this from historical records firstly moving down generations orally then by writting around 3500-4000 years ago when describing the looks of our earlier kings and leaders, all refer to them as having a lighter and whiter skin color, which if you looked into our mythology, you'd note that their skin color of being much lighter than the tone during the times of the recording being used to insinuate they were divine figures who were sent by the gods to lead the nation to salvation in the noble land (Arya, i.e the Iranian plateau) and that their skin color was like that because they "Shined" with divine spirit.
And since the first integration and recorded mixing process to start between us and the Roujams started around a 1000 years after we moved there, but yet we were still getting darker according to the stories means that the adoption process was beginning to happen even before mixing over the course of the first several thousands of years. Which is due to the radical difference in climate between the regions.
(edit: even if it wasn't genetic, the sun is enough to burn you
)
When the integration process really fully began ofcourse, the accelerated and now we look pretty identical to the more native population of the plateau.