TITLE: General distributions with the "transportation-cost inequality" property to piece log-concave distributions
QUESTION [5 upvotes]: It is now known [Otto et Villani 2000; Cordero et al 2006; etc.] that on an $n$-dimensional smooth Riemannian manifold $X$ and a probability measure $\mu$ on $X$ with density $d\mu \propto e^{-V}dvol$ satisfying the curvature condition 
$$
\operatorname{Hess}_x(V) + \operatorname{Ric}_x \succeq (1/c) I_n,\forall x \in X,
$$
the transportation-cost inequality
$$W_2(\nu,\mu) \le \sqrt{2cH(\nu\|\mu)}
$$
holds for all other distributions $\nu$ on $X$ absolutely continuous w.r.t $\mu$. Here $W_2$ is the Wasserstein-2 distance induced by the geodesic metric on $X$.
Question

(A) What are the most general conditions under which the above transportation-cost inequality holds.
(B) Can the curvature condition be relaxed to "piecewise" version. That is, what if we instead assume a convex mixture $d\mu = \sum_{i=1}^k \pi_i d\mu_i$ where each $d\mu_i$ is log-concave and satisfies the curvature condition on some piece $X_i$ of $X$ ?
(B') Are there concentration inequalities for mixtures of Gaussians ?


Partial answer

Log-concave distributions satisfying the Bakry-Eméry $\operatorname{CD}(n,\infty)$ curvature condition on manifolds
Distributions with densities on compact homogeneous Riemannian manifolds. See this paper of Rothaus.
Distributions which can be realized as pushforwards of distributions with some $\text{T}_2(c)$, under Lipschitz maps. If $\mu$ has $\text{T}_2(c)$ property and $\varphi: X \rightarrow Y$ is $L$-Lipschitz,  then $\varphi_\#\mu$ has $\text{T}_2(L^2c)$.
Finite tensor product  $\mu_1 \otimes \mu_2 \otimes \ldots \otimes \mu_k$ of distributions having $\text{T}_2(c)$ also has $\text{T}_2(c)$.

REPLY [2 votes]: (A) Nathael Gozlan proved that a distribution $\mu$ satisfies a $T_2$ inequality if and only if all finite tensor products of $\mu$ satisfy a subgaussian concentration property.
(B) and (B') For finite mixtures you could certainly get something by viewing them as pushforwards of product measures.  However, you would necessarily get something that depends badly on $k$, since any distribution on $\mathbb{R}^n$ --- including those with no good concentration properties --- can be approximated by a mixture of Gaussians.