We deal with kernel convergence of domains in ℂn which
are biholomorphically equivalent to the unit ball B. We also
prove that there is an equivalence between the convergence
on compact sets of biholomorphic mappings on B, which satisfy a growth theorem, and the kernel convergence. Moreover, we obtain certain consequences of this equivalence in the study of
Loewner chains and of starlike and convex mappings on B.