Let $M$ be a compact connected Riemannian manifold possibly with a boundary, let $V\in C^2(M)$ such that $μ(d x):=e^{V(x)}d x$ is a probability measure, and let $\{λ_i\}_{i\ge 1} $ be all non-trivial eigenvalues of $-L$ with Neumann boundary condition if the boundary exists. Then the empirical measures $\{μ_t\}_{t>0}$ of the diffusion process generated by $L$ (with reflecting boundary if the boundary exists) satisfy $$ \lim_{t\to \infty} \big\{t \mathbb E^x [W_2(μ_{t},μ)^2]\big\}= \sum_{i=1}^\infty\frac 2 {λ_i^2}\ \text{ uniformly\ in\ } x\in M,$$ where $\mathbb E^x$ denotes the expectation for the diffusion process starting at point $x$, $W_2$ is the $L^2$-Warsserstein distance induced by the Riemannian metric. The limit is finite if and only if $d\le 3$, and in this case we derive the following central limit theorem: $$\lim_{t\to\infty} \sup_{x\in M} \Big|\mathbb P^x(t W_2(μ_{t},μ)^2<a)- \mathbb P\Big( \sum_{k=1}^\infty \frac{2ξ_k^2}{λ_k^2}<a\Big)\Big|=0, \ \ a\ge 0,$$ where $\mathbb P^x$ is the probability with respect to $\mathbb E^x$, and $\{ξ_k\}_{k\ge 1}$ are i.i.d. standard Gaussian random variables. Moreover, when $d\ge 4$ we prove that the main order of $\mathbb E^x[W_2(μ_{t},μ)^2]$ is $t^{-\frac 2 {d-2}}$ as $t\to\infty$. Moreover, when $d\ge 4$ the main order of $\mathbb E^x[W_2(μ_{t},μ)^2]$ is $t^{-\frac 2 {d-2}}$ as $t\to\infty$. Finally, we establish the long-time large deviation principle for $\{W_2(μ_t,μ)^2\}_{t\ge 0}$ with a good rate function given by the information with respect to $μ$.