Suppose $X$ and $Y$ are smooth connected schemes over a field $k=\bar{k}$, $f: X\times_kY\to X$ is the first projection. You may assume $Y$ is proper if you like, my question is if $P\to X\times_kY$ is a $G$-torsor with $G$ finite group scheme over $k$ then can we say that the fibres of $P$ under $f$ are "constant". More precisely is there a functorial (with respect to $P$) isomorphism between $P_s$ and $P_{s'}$ for $s,s'$ any two rational points in $X$.

The similar question is that if $V$ is a vector bundle on $X\times_kY$ and if the fibres of $V$ along closed points of $X$ are all trivial vector bundles on $Y$, then do we have (in a functorial way) a vector bundle $W$ on $X$ such that $f^*W\cong V$ ?

The answer to the sencond problem is yes if we assume $Y$ is proper, this is a very standard algebraic geometry argument: using Grauert's base change theorem one can show that $f^*f_*V\to V$ is always an isomorphism. But what happens when $Y$ is not proper? do we have counter examples?