  This paper describes the quality of convergence to an infinitely divisible
law relative to free multiplicative convolution. We show that convergence in
distribution for products of identically distributed and infinitesimal free
random variables implies superconvergence of their probability densities to the
density of the limit law. Superconvergence to the marginal law of free
multiplicative Brownian motion at a specified time is also studied. In the
unitary case, the superconvergence to free Brownian motion and that to the Haar
measure are shown to be uniform over the entire unit circle, implying further a
free entropic limit theorem and a universality result for unitary free L\'{e}vy
processes. Finally, the method of proofs on the positive half-line gives rise
to a new multiplicative Boolean to free Bercovici-Pata bijection.
