  We introduce stochastic Interaction-Round-a-Face (IRF) models that are
related to representations of the elliptic quantum group $E_{\tau,\eta}(sl_2)$.
For stochasic IRF models in a quadrant, we evaluate averages for a broad family
of observables that can be viewed as higher analogs of $q$-moments of the
height function for the stochastic (higher spin) six vertex models.
  In a certain limit, the stochastic IRF models degenerate to (1+1)d
interacting particle systems that we call dynamic ASEP and SSEP; their jump
rates depend on local values of the height function. For the step initial
condition, we evaluate averages of observables for them as well, and use those
to investigate one-point asymptotics of the dynamic SSEP.
  The construction and proofs are based on remarkable properties (branching and
Pieri rules, Cauchy identities) of a (seemingly new) family of symmetric
elliptic functions that arise as matrix elements in an infinite volume limit of
the algebraic Bethe ansatz for $E_{\tau,\eta}(sl_2)$.
