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Let $\{S_n, n\geq1\}$ be a random walk wih independent and identicallydistributed increments and let $\{g_n,n\geq1\}$ be a sequence of real numbers.Let $T_g$ denote the first time when $S_n$ leaves $(g_n,\infty)$. Assume thatthe random walk is oscillating and asymptotically stable, that is, there existsa sequence $\{c_n,n\geq1\}$ such that $S_n/c_n$ converges to a stable law. Inthis paper we determine the tail behaviour of $T_g$ for all oscillatingasymptotically stable walks and all boundary sequences satisfying $g_n=o(c_n)$.Furthermore, we prove that the rescaled random walk conditioned to stay abovethe boundary up to time $n$ converges, as $n\to\infty$, towards the stablemeander.

First-passage times over moving boundaries for asymptotically stable walks