By using the spectrum of the underlying symmetric diffusion operator, the convergence in $L^p$-Wasserstein distance $\mathbb W_p (p\ge 1)$ is characterized for the empirical measure $μ_t$ of non-symmetric subordinated diffusion processes in an abstract framework. The main results are applied to the subordinations of several typical models, which include the (reflecting) diffusion processes on compact manifolds, the conditional diffusion processes, the Wright-Fisher diffusion process, and hypoelliptic diffusion processes on {\bf SU}(2). In particular, for the (reflecting) diffusion processes on a compact Riemannian manifold with invariant probability measure $μ$: (1) the sharp limit of $t\mathbb W_2(μ_t,μ)^2$ is derived in $L^q(\mathbb P)$ for concrete $q\ge 1,$ which provides a precise characterization on the physical observation that a divergence-free perturbation accelerates the convergence in $\mathbb W_2$; (2) the sharp convergence rates are presented for $(\mathbb E[\mathbb W_{2p}(μ_t,μ)^{q}])^{\frac 1 q} (p,q\ge 1)$, where a critical phenomenon appears with the critical rate $t^{-1}\log t$ as $t\to\infty$.