  We study a Lepton-Flavored Dark Matter model and its signatures at a future
Muon Collider. We focus on the less-explored regime of feeble dark matter
interactions, which suppresses the dangerous lepton-flavor violating processes,
gives rise to dark matter freeze-in production, and leads to long-lived
particle signatures at colliders. We find that the interplay of dark matter
freeze-in and its mediator freeze-out gives rise to an upper bound of around
TeV scales on the dark matter mass. The signatures of this model depend on the
lifetime of the mediator, and can range from generic prompt decays to more
exotic long-lived particle signals. In the prompt region, we calculate the
signal yield, study useful kinematics cuts, and report tolerable systematics
that would allow for a $5\sigma$ discovery. In the long-lived region, we
calculate the number of charged tracks and displaced lepton signals of our
model in different parts of the detector, and uncover kinematic features that
can be used for background rejection. We show that, unlike in hadron colliders,
multiple production channels contribute significantly which leads to sharply
distinct kinematics for electroweakly-charged long-lived particle signals.
Ultimately, the collider signatures of this lepton-flavored dark matter model
are common amongst models of electroweak-charged new physics, rendering this
model a useful and broadly applicable benchmark model for future Muon Collider
studies that can help inform work on detector design and studies of
systematics.
