A flow of sub-manifolds

Question

Let $(M,g)$ be a riemannian manifold and $f\in{\cal C}^{\infty}(M)$ a smooth function over $M$, a flow over sub-manifolds $X$ is defined by:

$$\frac{\partial X}{\partial t}=-grad (F)_X$$

where $F$ is the following functional:

$$F(X)=\int_X f(x) dx$$

Is this flow well defined? Have we:

$$grad(F)_X=(df)^*_X$$

?