  This paper deals with ergodic theorems for particular time-inhomogeneous
Markov processes, whose the time-inhomogeneity is asymptotically periodic.
Under a Lyapunov/minorization condition, it is shown that, for any measurable
bounded function $f$, the time average $\frac{1}{t} \int_0^t f(X_s)ds$
converges in $\mathbb{L}^2$ towards a limiting distribution, starting from any
initial distribution for the process $(X_t)_{t \geq 0}$. This convergence can
be improved to an almost sure convergence under an additional assumption on the
initial measure. This result will be then applied to show the existence of a
quasi-ergodic distribution for processes absorbed by an asymptotically periodic
moving boundary, satisfying a conditional Doeblin's condition.
