Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite $R$-modules, $p$ a prime ideal, and $I$ an ideal such that $IM\neq M$.

Definition: The common length of the maximal $M$-sequences in $I$ is called the grade of $I$ on $M$; denoted by $\operatorname{grade}(I,M)$.

$\operatorname{grade}(m,M)$ is denoted by $\operatorname{depth}M$. So by $\operatorname{depth}M_p$, we mean $\operatorname{grade}(pR_p,M_p)$.

Assuming $\operatorname{depth}M\ge \operatorname{depth}N$, what can one say about $\operatorname{depth}M_p$ and $\operatorname{depth}N_p$? Is there any inequality between them?

What if we impose additional assumptions on $M$ and $N$? For example, if $M=R/I$ and $N=R/J$?

Thank you.