  We study a multispecies $t$-PushTASEP system on a finite ring of $n$ sites
with site-dependent rates $x_1,\dots,x_n$. Let
$\lambda=(\lambda_1,\dots,\lambda_n)$ be a partition whose parts represent the
species of the $n$ particles on the ring. We show that for each composition
$\eta$ obtained by permuting the parts of $\lambda$, the stationary probability
of being in state $\eta$ is proportional to the ASEP polynomial
$F_{\eta}(x_1,\dots,x_n; q,t)$ at $q=1$; the normalizing constant (or partition
function) is the Macdonald polynomial $P_{\lambda}(x_1,\dots,x_n;q,t)$ at
$q=1$. Our approach involves new relations between the families of ASEP
polynomials and of non-symmetric Macdonald polynomials at $q=1$. We also use
multiline diagrams, showing that a single jump of the PushTASEP system is
closely related to the operation of moving from one line to the next in a
multiline diagram. We derive symmetry properties for the system under
permutation of its jump rates, as well as a formula for the current of a
single-species system.
