# How to understand the flow of time-depended vector field

Maybe, this question shows my stupidity. When I read Amitai Yuval's answer, I can't understand it. In fact, I only know what is the flow of a fixed vector field, fixed means it does not depend of time. Besides, since the flow of vector field is defined by integral curve, I don't know how to calculate $\partial_t \varphi$. So I really don't know how to verify $F'_t$ is mean curvature flow in picture below.

Besides, as I know , only flow of vector field with compact support is defined on all $t\in (-\infty, +\infty)$. Why compactness of $M$ makes the flow can be defined on all $M\times[0,T)$. I know the compactness of $M$ means the vector field has compact support, but I don't know $t\in [0,T)$ is is parameter of flow or time.