$\lim\limits_{k\to\infty}F_{\varepsilon_k}(x_{\varepsilon_k})=\liminf\limits_{\varepsilon\to0}\inf\limits_{x\in K}F_{\varepsilon}(x)$?
If $A$ denotes the right side then we can write $A=\lim_k (\inf_{x \in K} F_{\epsilon_k} (x))$ for some sequence $(\epsilon_k)$ decreasing to $0$. For each $k$ we can find $x_k \in K$ such that $F_{\epsilon_k} (x_k)-\frac 1 k <\inf_{x \in K} F_{\epsilon_k} (x)) \leq F_{\epsilon_k} (x_k)$. Now $F_{\epsilon_k} (x_k)$ tends to $A$.