  We prove that Broder's Markov chain for approximate sampling near-perfect and
perfect matchings is not rapidly mixing for Hamiltonian, regular, threshold and
planar bipartite graphs, filling a gap in the literature. In the second part we
experimentally compare Broder's chain with the Markov chain by Jerrum, Sinclair
and Vigoda from 2004. For the first time, we provide a systematic experimental
investigation of mixing time bounds for these Markov chains. We observe that
the exact total mixing time is in many cases significantly lower than known
upper bounds using canonical path or multicommodity flow methods, even if the
structure of an underlying state graph is known. In contrast we observe
comparatively tighter upper bounds using spectral gaps.
