  Let $G$ be a simple graph of order $n$ and let $k$ be an integer such that
$1\leq k\leq n-1$. The $k$-token graph $G^{\{k\}}$ of $G$ is the graph whose
vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in
$G^{\{k\}}$ whenever their symmetric difference is a pair of adjacent vertices
in $G$. In this paper we study the Hamiltonicity of the $k$-token graphs of
some join graphs. As a consequence, we provide an infinite family of graphs
(containing Hamiltonian and non-Hamiltonian graphs) for which their $k$-token
graphs are Hamiltonian. Our result provides, to our knowledge, the first family
of non-Hamiltonian graphs for which their $k$-token graphs are Hamiltonian, for
$2<k<n-2$.
