Let Mp(X,T)\documentclass[12pt]{minimal}
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\begin{document}$$M_p(X,T)$$\end{document} denote the Markov type p constant at time T of a metric space X, where p≥1\documentclass[12pt]{minimal}
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\begin{document}$$p \ge 1$$\end{document}. We show that Mp(Y,T)≤Mp(X,T)\documentclass[12pt]{minimal}
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\begin{document}$$M_p(Y,T) \le M_p(X,T)$$\end{document} in each of the following cases: (a) X and Y are geodesic spaces and Y is covered by X via a finite-sheeted locally isometric covering, (b) Y is the quotient of X by a finite group of isometries, (c) Y is the Lp\documentclass[12pt]{minimal}
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\begin{document}$$L^p$$\end{document}-Wasserstein space over X. As an application of (a) we show that all compact flat manifolds have Markov type 2 with constant 1. In particular the circle with its intrinsic metric has Markov type 2 with constant 1. This answers the question raised by S.-I. Ohta and M. Pichot. Parts (b) and (c) imply new upper bounds for Markov type constants of the Lp\documentclass[12pt]{minimal}
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\begin{document}$$L^p$$\end{document}-Wasserstein space over Rd\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^d$$\end{document}. These bounds were conjectured by A. Andoni, A. Naor and O. Neiman. They imply certain restrictions on bi-Lipschitz embeddability of snowflakes into such Wasserstein spaces.
