  We study a class of almost diagonal matrices compatible with the mixed-norm
$\alpha$-modulation spaces $M_{\vec{p},q}^{s,\alpha}(\mathbb{R}^n)$, $\alpha\in
[0,1]$, introduced recently by Cleanthous and Georgiadis [Trans.\ Amer.\ Math.\
Soc.\ 373 (2020), no. 5, 3323-3356]. The class of almost diagonal matrices is
shown to be closed under matrix multiplication and we connect the theory to the
continuous case by identifying a suitable notion of molecules for the
mixed-norm $\alpha$-modulation spaces. We show that the "change of frame"
matrix for a pair of time-frequency frames for the mixed-norm
$\alpha$-modulation consisting of suitable molecules is almost diagonal. As
examples of applications, we use the almost diagonal matrices to construct
compactly supported frames for the mixed-norm $\alpha$-modulation spaces, and
to obtain a straightforward boundedness result for Fourier multipliers on the
mixed-norm $\alpha$-modulation spaces.
