During the last few decades cultural changes have been taking place in many countries due to migration. The degree to which the foreign culture influences the local culture, differs across countries. This paper shows how the willingness of locals and immigrants to intermarry influences the culture and the national identity of the host country. We use a search-theoretic approach to show that, even in situations where migrants and natives prefer to marry within their own community, the search process may lead to intermarriage. The exogamy can take on two forms: either migrants and natives each hold on to their own culture or the immigrants take on the natives' culture. In the first case we will see new cultures developing and the local culture will not survive over time. In the second case the local culture will survive. We show the conditions for assimilation versus no assimilation between the groups.