  We present theoretical predictions for $\mu \rightarrow e$ conversion rates
using a tower of effective field theories connecting the UV to nuclear physics
scales. The interactions in nuclei are described using a recently developed
nonrelativistic effective theory (NRET) that organizes contributions according
to bound nucleon and muon velocities, $\vec{v}_N$ and $\vec{v}_\mu$, with
$|\vec{v}_N| > |\vec{v}_\mu|$. To facilitate the top-down matching, we enlarge
the set of Lorentz covariant nucleon-level interactions mapped onto the NRET
operators to include those mediated by tensor interactions, in addition to the
scalar and vector interactions already considered previously, and then match
NRET nonperturbatively onto the Weak Effective Theory (WET). At the scale $\mu
\approx 2$ GeV WET is formulated in terms of $u$, $d$, $s$ quarks, gluons and
photons as the light degrees of freedom, along with the flavor-violating
leptonic current. We retain contributions from WET operators up to dimension 7,
which requires the full set of 26 NRET operators. The results are encoded in
the open-source Python- and Mathematica-based software suite MuonBridge, which
we make available to the theoretical and experimental communities interested in
$\mu \rightarrow e$ conversion.
