Parallel Transport & Covariant derivatives

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melinoe<3
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Hello! I've been having trouble with a problem regarding parallel transport. Here is essentially the problem: Suppose I have a vector X on a manifold M with a connection $\Delta$. $\Delta_X$ is the vovariant derivative along X. Also, in this problem we have a differential operator $exp(-t \Delta_X)$. And we define notation such that $X|_P$ is the vector at point P $\in$ M.

If I introduce another vector field W, we can define $W_t = exp(-t \Delta_X)W$, a family of vector fields. If we consider a flow curve $\gamma_t (P)$ of X, where
 $\gamma_0 (P) = P$ and $\dot{\gamma_t} (P) = X|_{\gamma_t (P)}$, the dot being the time derivative. On the flow curve, define $w(t) =
W_t |_{\gamma_t (P)} \in TM_{\gamma_t (P)}$. I want to show that w(t) is transported along $\gamma_t (P)$.

I'm stuck on a few things. First, how should I interpret the differential operator? Since the covariant derivatuve acts locally, is it okay to use the taylor expansion of $e^x$ up to first order. Second, I'm not sure how to define the covariant derivative along a curve. I understand that the tangent vector X is equal to $\dot{\gamma}$ but then what? Also I'm having trouble showing how the parallel transport happens.