  An infinite-type surface $\Sigma$ is of type $\mathcal{S}$ if it has an
isolated puncture $p$ and admits shift maps. This includes all infinite-type
surfaces with an isolated puncture outside of two sporadic classes. Given such
a surface, we construct an infinite family of intrinsically infinite-type
mapping classes that act loxodromically on the relative arc graph
$\mathcal{A}(\Sigma, p)$. J. Bavard produced such an element for the plane
minus a Cantor set, and our result gives the first examples of such mapping
classes for all other surfaces of type $\mathcal{S}$. The elements we construct
are the composition of three shift maps on $\Sigma$, and we give an alternate
characterization of these elements as a composition of a pseudo-Anosov on a
finite-type subsurface of $\Sigma$ and a standard shift map. We then explicitly
find their limit points on the boundary of $\mathcal{A}(\Sigma,p)$ and their
limiting geodesic laminations. Finally, we show that these infinite-type
elements can be used to prove that Map$(\Sigma,p)$ has an infinite-dimensional
space of quasimorphisms.
