  Let $Z_N$ be a Ginibre ensemble and let $A_N$ be a Hermitian random matrix
independent from $Z_N$ such that $A_N$ converges in distribution to a
self-adjoint random variable $x_0$. For each $t>0$, the random matrix
$A_N+\sqrt{t}Z_N$ converges in $\ast$-distribution to $x_0+c_t$, where $c_t$ is
the circular variable of variance $t$, free from $x_0$. We use the
Hamilton-Jacobi method to compute the Brown measure $\rho_t$ of $x_0+c_t$. The
Brown measure has a density that is constant along the vertical direction
inside the support. The support of the Brown measure of $x_0+c_t$ is related to
the subordination function of the free additive convolution of $x_0+s_t$, where
$s_t$ is the semicircular variable of variance $t$, free from $x_0$.
Furthermore, the push-forward of $\rho_t$ by a natural map is the law of
$x_0+s_t$.
  Let $G_N(t)$ be the Brownian motion on the general linear group and let $U_N$
be a unitary random matrix independent from $G_N(t)$ such that $U_N$ converges
in distribution to a unitary random variable $u$. The random matrix $U_NG_N(t)$
converges in $\ast$-distribution to $ub_t$ where $b_t$ is the free
multiplicative Brownian motion, free from $u$. We compute the Brown measure
$\mu_t$ of $ub_t$, extending the recent work by Driver-Hall-Kemp, which
corresponds to the case $u=I$. The measure has a density of the special form
  \[\frac{1}{r^2}w_t(\theta)\] in polar coordinates in its support. The support
of $\mu_t$ is related to the subordination function of the free multiplicative
convolution of $uu_t$ where $u_t$ is the free unitary Brownian motion, free
from $u$. The push-forward of $\mu_t$ by a natural map is the law of $uu_t$. We
compute the explicit formula for the special case where $u$ is Haar unitary.
The support of the Brown measure of $ub_t$ is an annulus; in its support, the
density in polar coordinates is given by
  \[\frac{1}{2\pi t}\frac{1}{r^2}.\]
