We study the complex-time Segal-Bargmann transform
$\mathbf{B}_{s,\tau}^{K_N}$ on a compact type Lie group $K_N$, where $K_N$ is
one of the following classical matrix Lie groups: the special orthogonal group
$\mathrm{SO}(N,\mathbb{R})$, the special unitary group $\mathrm{SU}(N)$, or the
compact symplectic group $\mathrm{Sp}(N)$. Our work complements and extends the
results of Driver, Hall, and Kemp on the Segal-Bargman transform for the
unitary group $\mathrm{U}(N)$. We provide an effective method of computing the
action of the Segal-Bargmann transform on \emph{trace polynomials}, which
comprise a subspace of smooth functions on $K_N$ extending the polynomial
functional calculus. Using these results, we show that as $N\to\infty$, the
finite-dimensional transform $\mathbf{B}_{s,\tau}^{K_N}$ has a meaningful limit
$\mathscr{G}_{s,\tau}^{(\beta)}$ (where $\beta$ is a parameter associated with
$\mathrm{SO}(N,\mathbb{R})$, $\mathrm{SU}(N)$, or $\mathrm{Sp}(N)$), which can
be identified as an operator on the space of complex Laurent polynomials