  Bilayer graphene in a perpendicular electric field can host domain walls
between regions of reversed field direction or interlayer stacking. The gapless
modes propagating along these domain walls, while not strictly topological,
nevertheless have interesting physical properties, including valley-momentum
locking. A junction where two domain walls intersect forms the analogue of a
quantum point contact. We study theoretically the critical behavior of this
junction near the pinch-off transition, which is controlled by two separate
classes of non-trivial quantum critical points. For strong interactions, the
junction can host phases of unique charge and valley conductances. For weaker
interactions, the low-temperature charge conductance can undergo one of two
possible quantum phase transitions, each characterized by a specific critical
exponent and a collapse to a universal scaling function, which we compute.
