  We consider a family of free multiplicative Brownian motions $b_{s,\tau}$
parametrized by a positive real number $s$ and a nonzero complex number $\tau$
satisfying $\left\vert \tau-s\right\vert \leq s,$ with an arbitrary unitary
initial condition. We compute the Brown measure $\mu_{s,\tau}$ of $b_{s,\tau}$
and find that it has a simple structure, with a density in logarithmic
coordinates that is constant in the $\tau$-direction. We also find that all the
Brown measures with $s$ fixed and $\tau$ varying are related by pushforward
under a natural family of maps. Our results generalize those of
Driver-Hall-Kemp and Ho-Zhong for the case $\tau=s.$ We use a version of the
PDE method introduced by Driver-Hall-Kemp, but with some significant technical
differences.
