score:7
One term to describe this is cultural diffusion.
"The spreading out of culture...from a central point."
Today's "central point" might be America or China; Britain or France during the Industrial Revolution, and Greece or Rome in ancient times.
Upvote:4
One term that comes to mind is Acculturation. You mentioned a larger culture influencing smaller surrounding cultures and that is more or less the definition of acculturation. It's not necessarily a forced effect, but rather through contact between the cultures, the larger culture tends to "assimilate" the less dominant cultures into it.