Let $I, J$ be a pair of ideals of a Noetherian local ring $R$. The purpose of this paper is to study asymptotic properties such as the asymptotic stability of the depths in dimension $> k$, $\mathrm{depth}_k(I, M_n)$ and $\mathrm{depth}_k(I,M_n/M_{n+1})$ of an ideal $I$ with respect to a $J$-filtration $(M_n)$ of submodules of a finitely generated module $M$; or the asymptotic behaviour of sets of associated prime ideals or of attached prime ideals of certain local cohomology modules determined by $M$ and $I, J$.