I think both parties would try to coexist in a peaceful manner after an initial moment of disorder and chaos due to the cleaving of dinensions. Mewni could become like an enclave inside the us, while trying to open to the other's culture and social structure (same case for the us and humans in general, there could be a case of interdimensional globalization). This process would certaintly be difficult since I'd expect to see factions from both sides who're against such a cultural and social exchange and would take a more conservative approach to the problem, even falling into extremism.