The second paper in this series is devoted to the convergence of sequences of infinite determinantal measures, understood as the convergence of sequences of the corresponding finite determinantal measures. Besides the weak topology in the space of probability measures on the space of configurations, we also consider the natural immersion (defined almost surely with respect to the infinite Bessel process) of the space of configurations into the space of finite measures on the half-line, which induces a weak topology in the space of finite measures on the space of finite measures on the half-line. The main results of the present paper are sufficient conditions for the tightness of families and the convergence of sequences of induced determinantal processes as well as for the convergence of processes corresponding to finite-rank perturbations of operators.