There is no easy answer... You will have to check out cemplexation constants for CaClx, ZnClx, Ca(CO3)x, Zn(CO3)x and solubility proucts for ZnCO3 and CaCO3. Not to mention CO3(2-) hydrolysis and OH- complexes...
Basically ZnCO3 is about 100 times less soluble then CaCO3 so you may expect something like
ZnCl2(aq) + CaCO3(s) -> ZnCO3(s) + CaCl2
but it will be very slow and difficult to spot, as the solution will look all the same, with white precipitate.