Every more successful civilization will replace a less successful civilization. In the past it was done by horrible means - initially though genocide, then through enslavement, later through foreign rule that was a bit less terrible for an average resident. Today, it's through economic power, aka gentrification. Visit Silicon Valley if you have any questions.

As the means become less brutal, we should perhaps welcome still inevitable outcomes. If populations of Canary Islands had a chance to evaluate Spanish civilization through entirely voluntary means, more and more would choose to adopt a relatively comfortable lifestyle rather than constant struggle with scarcity. And those who do not have to accept that they are practicing a hobby that will gather a limited number of adherents. So long as your individual rights are respected and your individual responsibility for outcomes is emphasized, I don't see a problem with dominant culture changing over time.