  Let $M_p(X,T)$ denote the Markov type $p$ constant at time $T$ of a metric
space $X$, where $p \ge 1$. We show that $M_p(Y,T) \le M_p(X,T)$ 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 $\mathbb R^d$. 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.
