For entourage $U$ and continuous map $t:X\to X$, is there an entourage $V$ with $t^{-1}V[x]\subseteq U[x]$?

Question

Let $(X, \mathcal{U})$ be a uniform space, $T$ be a topological semigroup and $(T, X)$ be a semiflow. This means that $t:X\to X$ is continuous and $(t_0t_1)(x)=t_0(t_1x)$ and if $e\in T$, then $ex=x$. For $A\subseteq T$ and $x\in X$, define $A^{-1}x=\{y: \text{ there is } a\in A \text{ such that } ay=x\}$. Consider an entourage $U\in\mathcal{U}$ and compact set $K$ in $T$. Is there an entourage $V\in\mathcal{U}$ such that $K^{-1}V[x]\subseteq U[x]$ for all $x\in X$.