I just remembered that there might another way using Jacobi fields. Suppose you have a hypersurface in an $(n+1)$-dimensional Riemannian manifold. You take advantage of the fact that along each geodesic normal to an $n$-dimensional hypersurface, there are $n$ linearly independent Jacobi fields $J_1, \dots, J_n$ that are tangent to the level hypersurfaces of the distance from the original hypersurface, where $g_{ij} = J_i\cdot J_j$ is the induced metric of the level hypersurface at each time $t$. I believe that you can write down a formula for the second fundamental form for each level hypersurface in terms of the Jacobi fields and their covariant derivatives (including with respect to $t$). From there you can compute its first variation.

A good paper to study is the classic by Heintze and Karcher.

I just remembered that there might another way using Jacobi fields. Suppose you have a hypersurface in an $(n+1)$-dimensional Riemannian manifold. You take advantage of the fact that along each geodesic normal to an $n$-dimensional hypersurface, there are $n$ linearly independent Jacobi fields $J_1, \dots, J_n$ that are tangent to the level hypersurfaces of the distance from the original hypersurface, where $g_{ij} = J_i\cdot J_j$ is the induced metric of the level hypersurface at each time $t$. The second fundamental form for each level hypersurface is equal to $h_{ij} = J_i\cdot\nabla_t J_j$. From there you can compute its first variation.

A good paper to study is the classic by Heintze and Karcher.