  We study the role of group infall in the assembly and dynamics of galaxy
clusters in $\Lambda$CDM. We select $10$ clusters with virial mass $M_{\rm 200}
\sim 10^{14} \, M_{\odot}$ from the cosmological hydrodynamical simulation
Illustris and follow their galaxies with stellar mass $M_{\star} \geq 1.5
\times 10^8 \, M_{\odot}$. A median of $\sim 38\%$ of surviving galaxies at
$z=0$ are accreted as part of groups and did not infall directly from the
field, albeit with significant cluster-to-cluster scatter. The evolution of
these galaxy associations is quick, with observational signatures of their
common origin eroding rapidly in $1$-$3$ Gyr after infall. Substructure plays a
dominant role in fostering the conditions for galaxy mergers to happen, even
within the cluster environment. Integrated over time, we identify (per cluster)
an average of $17 \pm 6$ mergers that occur in infalling galaxy associations,
of which $7 \pm 3$ occur well within the virial radius of their cluster hosts.
The number of mergers show large dispersion from cluster to cluster, with our
most massive system having $42$ mergers above our mass cut-off. These mergers,
which are typically gas rich for dwarfs and a combination of gas rich and gas
poor for $M_{\star} \sim 10^{11} \, M_{\odot}$, may contribute significantly
within $\Lambda$CDM to the formation of specific morphologies, such as
lenticulars (S0) and blue compact dwarfs in groups and clusters.
