Let $I$ be an ideal in $S=K[x_1,\dots,x_n]$. Can we compute $\operatorname{depth}(I\cap K[x_1,\dots,x_r])$ with $r \leq n$? Is there any relation between depth $I$ and $\operatorname{depth}(I\cap K[x_1,\dots,x_r])$?

And what can we tell about $\operatorname{depth}(M \cap N)$ when $M,N$ are $S$-modules? Does it have any bounds?