In the book `Optimal Transport for Applied Mathematicians' by Santambrogio, on page 289 it is written that 
In this setup, the space $X=(X,d)$ is a metric space, $\{\tilde{x}^\tau\}_{\tau>0}$ is a sequence such that for each $t \in[0,T]$, $\tilde{x}^\tau(t)\in X$, and $|(\tilde{x}^\tau)'|(t)$ is the metric derivative.
$\textbf{Question :}$ Does anyone know where I can find a reference of this specific embedding theorem?
EDIT : I realized the problem, if the underlying space $\Omega$ is assumed to be compact then compactness is straightforward (this is the case in Santambrogios book). If however we are working on a non-compact set, e.g. $\mathbb{R}^d$ then compactness with this method is only achieved in $W_p$ for $1\leq p <2$. The equicontinuity + the compactness give the uniform convergence in whatever space we have the compactness.