  Many integrable stochastic particle systems in one space dimension (such as
TASEP - totally asymmetric simple exclusion process - and its various
deformations, with a notable exception of ASEP) remain integrable when we equip
each particle $x_i$ with its own jump rate parameter $\nu_i$. It is a
consequence of integrability that the distribution of each particle $x_n(t)$ in
a system started from the step initial configuration depends on the parameters
$\nu_j$, $j\le n$, in a symmetric way. A transposition $\nu_n \leftrightarrow
\nu_{n+1}$ of the parameters thus affects only the distribution of $x_n(t)$.
For $q$-Hahn TASEP and its degenerations ($q$-TASEP and directed beta polymer)
we realize the transposition $\nu_n \leftrightarrow \nu_{n+1}$ as an explicit
Markov swap operator acting on the single particle $x_n(t)$. For beta polymer,
the swap operator can be interpreted as a simple modification of the lattice on
which the polymer is considered. Our main tools are Markov duality and contour
integral formulas for joint moments. In particular, our constructions lead to a
continuous time Markov process $\mathsf{Q}^{(\mathsf{t})}$ preserving the time
$\mathsf{t}$ distribution of the $q$-TASEP (with step initial configuration,
where $\mathsf{t}\in \mathbb{R}_{>0}$ is fixed). The dual system is a certain
transient modification of the stochastic $q$-Boson system. We identify
asymptotic survival probabilities of this transient process with $q$-moments of
the $q$-TASEP, and use this to show the convergence of the process
$\mathsf{Q}^{(\mathsf{t})}$ with arbitrary initial data to its stationary
distribution. Setting $q=0$, we recover the results about the usual TASEP
established recently in [arXiv:1907.09155] by a different approach based on
Gibbs ensembles of interlacing particles in two dimensions.
