  We study the asymptotic behaviour of a family of dynamic models of crawling
locomotion, with the aim of characterizing a gait as a limit property. The
locomotors, which might have a discrete or continuous body, move on a line with
a periodic prescribed shape change, and might possibly be subject to external
forcing (e.g., crawling on a slope). We discuss how their behaviour is affected
by different types of friction forces, including also set-valued ones such as
dry friction. We show that, under mild natural assumptions, the dynamics always
converge to a relative periodic solution. The asymptotic average velocity of
the crawler yet might still depend on its initial state, so we provide
additional assumption for its uniqueness. In particular, we show that the
asymptotic average velocity is unique both for strictly monotone friction
forces, and also for dry friction, provided in the latter case that the
actuation is sufficiently smooth (for discrete models) or that the friction
coefficients are always nonzero (for continuous models). We present several
examples and counterexamples illustrating the necessity of our assumptions.
