In $\mathbb{R}^n$, the material derivative along a divergence free vector filed $\mathbf{u}$ is given by $\frac{D}{dt}=\partial_t+\mathbf{u}\cdot\nabla$, and has the Lagrangian formulation
Let $\gamma:\mathscr{D}\to\mathbb{R}^n$ be the flow of $\mathbf{u}$ (where $\mathscr{D}\subset\mathbb{R}\times\mathbb{R}^n$ is its domain of definition). Then, for all vector fields $X$ $$\frac{D}{dt}X(t,\gamma(t,x))=\partial_tX(t,\gamma(t,x)),$$
i.e. the material derivative restricted to integral curves is the classical time derivative. My question is: what is the corresponding formulation on a Riemannian manofold $M$?
In this case, the material derivative is $\partial_t+\nabla_u$, and sends vector fields to vector fields. But something like $$\partial_t X(t,\gamma(t,x))$$ belongs to $TTM$.
The most straightforward way of generalizing the definition you provide to $\mathbb{R}\times M$ makes use of the Levy-Civita connection. Let $X$ be a time dependent vector field on $M$, and let $U$ be the flow field. We can define the material derivative as $$ \frac{D}{dt}X=\frac{d}{dt}X+\nabla_UX $$ Where $\nabla_U$ is the covariant derivative. The LC connection also gives us a way of interpreting the derivative of a vector field along a curve, which reproduces the expression you provide. $$ \left(\frac{D}{dt}X\right)(t,\gamma(x,t))=D_t(X(t,\gamma(x,t))) $$ Where $D_t$ is the covariant derivative along the path $\gamma$, which is an integral curve of $U$.