  We derive Wasserstein distance bounds between the probability distributions
of a stochastic integral (It\^o) process with jumps $(X_t)_{t\in [0,T]}$ and a
jump-diffusion process $(X^\ast_t)_{t\in [0,T]}$. Our bounds are expressed
using the stochastic characteristics of $(X_t)_{t\in [0,T]}$ and the
jump-diffusion coefficients of $(X^\ast_t)_{t\in [0,T]}$ evaluated in $X_t$,
and apply in particular to the case of different jump characteristics. Our
approach uses stochastic calculus arguments and $L^p$ integrability results for
the flow of stochastic differential equations with jumps, without relying on
the Stein equation.
