  Let $\mathcal{P}$ be a graph property which is preserved by removal of edges,
and consider the random graph process that starts with the empty $n$-vertex
graph and then adds edges one-by-one, each chosen uniformly at random subject
to the constraint that $\mathcal{P}$ is not violated. These types of random
processes have been the subject of extensive research over the last 20 years,
having striking applications in extremal combinatorics, and leading to the
discovery of important probabilistic tools. In this paper we consider the
$k$-matching-free process, where $\mathcal{P}$ is the property of not
containing a matching of size $k$. We are able to analyse the behaviour of this
process for a wide range of values of $k$; in particular we prove that if
$k=o(n)$ or if $n-2k=o(\sqrt{n}/\log n)$ then this process is likely to
terminate in a $k$-matching-free graph with the maximum possible number of
edges, as characterised by Erd\H{o}s and Gallai. We also show that these bounds
on $k$ are essentially best possible, and we make a first step towards
understanding the behaviour of the process in the intermediate regime.
