  In this manuscript, we provide a non-asymptotic process level control between
the telegraph process and the Brownian motion with suitable diffusivity
constant via a Wasserstein distance with quadratic average cost. In addition,
we derive non-asymptotic estimates for the corresponding time average $p$-th
moments. The proof relies on coupling techniques such as coin-flip coupling,
synchronous coupling and the Koml\'os--Major--Tusn\'ady coupling.
