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I guess you assume that $F_0$ is bounded(?). . In this case the answer is "yes".

Fix a point $p$. Fix small $\varepsilon>0$ smaller than convexity radiusso that principle curvatures of $\varepsilon$ spheres in $M$ are uniformly bounded. (It is possible since $M$ has bounded geometry.)

Denote by $S_r$ the sphere with center $p$ and radius $r$ and let $\Sigma_r$ be the inward $\varepsilon$-equidistant to $S_r$.

Note that for $r>2\cdot\varepsilon$ there is a fixed upper bound for principle curvatures of $\Sigma_r$. Therefore $\ell(t)=\max_{x\in F_t}\{\mathop{\rm dist}_px\}$ grows at most linearly.

P.S. I used that $M$ has bounded geometry in an essential way, but I am not sure if it is necessary.

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I guess you assume that $F_0$ is bounded (?). In this case the answer is "yes".

Fix a point $p$. Fix $\varepsilon>0$ smaller than convexity radius.

Denote by $S_r$ the sphere with center $p$ and radius $r$ and let $\Sigma_r$ be the inward $\varepsilon$-equidistant to $S_r$.

Note that for $r>2\cdot\varepsilon$ there is a fixed upper bound for principle curvatures of $\Sigma_r$. Therefore $\ell(t)=\max_{x\in F_t}\{\mathop{\rm dist}_px\}$ grows at most linearly.