Let M_p(X, T) denote the Markov type p constant at time T of a metric space X, where p≥ 1. We show that M_p(Y, T) ≤ M_p(X, T) 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 L^p-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 L^p-Wasserstein space over R^d. 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.