  If $M$ is an atomic monoid and $x$ is a nonzero non-unit element of $M$, then
the set of lengths $\mathsf{L}(x)$ of $x$ is the set of all possible lengths of
factorizations of $x$, where the length of a factorization is the number of
irreducible factors (counting repetitions). In a recent paper, F. Gotti and C.
O'Neil studied the sets of elasticities $\mathcal{R}(P) := \{\sup
\mathsf{L}(x)/\inf \mathsf{L}(x) : x \in P\}$ of Puiseux monoids $P$. Here we
take this study a step forward and explore the local $k$-elasticities of the
same class of monoids. We find conditions under which Puiseux monoids have all
their local elasticities finite as well as conditions under which they have
infinite local $k$-elasticities for sufficiently large $k$. Finally, we focus
our study of the $k$-elasticities on the class of primary Puiseux monoids,
proving that they have finite local $k$-elasticities if either they are
boundedly generated and do not have any stable atoms or if they do not contain
$0$ as a limit point.
