When we can permute between the integral and convex hull?

Question

Is there a relation between the following expressions?

$$\operatorname{conv}\left(\int_{0}^{t} f(s,x)ds :x \in A \right) $$ and $$\int_{0}^{t} \operatorname{conv}(f(s,x):x \in A)\ ds $$ where $A$ is a bounded set and $\operatorname{conv}(A)$ is the convex hull of a set $A$.