  We present a general approach to study a class of random growth models in
$n$-dimensional Euclidean space. These models are designed to capture basic
growth features which are expected to manifest at the mesoscopic level for
several classical self-interacting processes originally defined at the
microscopic scale. It includes once-reinforced random walk with strong
reinforcement, origin-excited random walk, and few others, for which the set of
visited vertices is expected to form a "limiting shape". We prove an averaging
principle that leads to such shape theorem. The limiting shape can be computed
in terms of the invariant measure of an associated Markov chain.
