We investigate within a model of cultural transmission the conditions under which increased social diversity within a population - e.g. due to the inflow of immigrants - raise the potential for conflict as opposed to harmonious social diversity. Drawing on evidence from psychological studies, we develop the concept of "bridging identity", an individual trait that (i) directly affects utility in culturally diverse social groups but is immaterial in culturally homogeneous social groups; (ii) is fostered (probabilistically) in those born in culturally diverse social groups but not in those born in culturally homogeneous social groups. We find first, increased cultural diversity within a population can lead to more mixed social groups or increased segregation depending on the paceof change. This is in contrast to Schelling's models of residential segregation which would always predict increased segregation. Furthermore, a temporary negative shock to bridging identity can trigger a dynamic process of segregation in the form of outmigration from culturally diverse social groups. But, paradoxically, if the shock is severe enough, its effects are mitigated.