Union of Associated Primes being finite.

Question

Let $R$ be a commutative Noetherian ring with unit. Let $I=(x_1,x_2,\dots,x_t)$ be a nonzero ideal of $R$. Define $I_n=(x_1^n, x_2^n,...,x_t^n)$.

1. Are there known results about $\cup_n \operatorname{Ass}(R/I_n)$ being finite?
2. More generally, let $M$ be a finitely generated $R$-module. Are there any results about $\cup_n\operatorname{Ass}(M/I_nM)$ being finite?

Added Later: Is something known about $\cup_n\operatorname{Ass}(I^n/I_n)$?