  Motivated by various applications, we describe the scaling limits of
bivariate Markov chains $(X,J)$ on $\mathbb Z_+ \times \{1,\ldots,\kappa\}$
where $X$ can be viewed as a position marginal and $\{1,\ldots,\kappa\}$ is a
set of $\kappa$ types. The chain starts from an initial value $(n,i)\in \mathbb
Z_+ \times \{1,\ldots,\kappa\}$, with $i$ fixed and $n \rightarrow \infty$, and
typically we will assume that the macroscopic jumps of the marginal $X$ are
rare, i.e. arrive with a probability proportional to a negative power of the
current state. We also assume that $X$ is non-increasing. We then observe
different asymptotic regimes according to whether the rate of type change is
proportional to, faster than, or slower than the macroscopic jump rate. In
these different situations, we obtain in the scaling limit Lamperti transforms
of Markov additive processes, that sometimes reduce to standard positive
self-similar Markov processes. As first examples of applications, we study the
number of collisions in coalescents in varying environment and the scaling
limits of Markov random walks with a barrier. This completes previous results
obtained by Haas and Miermont as well as Bertoin and Kortchemski in the
monotype setting. In a companion paper, we will use these results as a building
block to study the scaling limits of multi-type Markov branching trees, with
applications to growing models of random trees and multi-type Galton-Watson
trees.
