Different equations for mean curvature flow

Question

Let $M$ be a manifold. Then a smooth family of immersions $F:M \times [0,T) \rightarrow \mathbb{R}^m$ is said to be a mean curvature flow if $$ \frac{\partial F}{\partial t} = \vec{H}, \: \: \: \: \: (1) $$ where $\vec{H}$ is the mean curvature vector of $M_t = F(M,t)$.
If one is only interested in the sets $(M_t)_{t \in [0,T)}$, it is common to say that $F$ is a mean curvature flow if it only satisfies $$ \left( \frac{\partial F}{\partial t} \right)^\perp = \vec{H}. \: \: \: \: \: (2)$$ I know that if (2) holds when $M$ is compact, then there exists a smooth family of diffeomorphisms $(\varphi_t)_{t \in [0,T)}$ of $M$ such that $\tilde{F}(p,t) = F(\varphi_t(p),t)$ satisfies (1).
However if $M$ is not compact such a family is only garanteed to exist locally around every point $p$ in $M$ and around every time $t$.

I would like to know if there are examples of smooth families for which (2) holds but for which no global smooth family of diffeomorphisms exists such that (1) holds after composition.

Secondly if an answer to the first question isn't known, are there examples when (2) holds, but when there exists no smooth family $G:M \times [0,T) \rightarrow \mathbb{R}^m$ such that $G$ satisfies (1) and $G(M,t) = F(M,t)$?