  We define interacting particle systems on configurations of the integer
lattice (with values in some finite alphabet) by the superimposition of two
dynamics: a substitution process with finite range rates, and a circular
permutation mechanism(called "cut-and-paste") with possibly unbounded range.
The model is motivated by the dynamics of DNA sequences: we consider an ergodic
model for substitutions, the RN+YpR model ([BGP08]), with three particular
cases, the models JC+cpg,T92+cpg, and RNc+YpR. We investigate whether they
remain ergodic with the additional cut-and-paste mechanism, which models
insertions and deletions of nucleotides. Using either duality or attractiveness
techniques, we provide various sets of sufficient conditions, concerning only
the substitution rates, for ergodicity of the superimposed process. They imply
ergodicity of the models JC+cpg, T92+cpg as well as the attractive RNc+YpR, all
with an additional cut-and-paste mechanism.
