We consider the numerical solution of the optimal transport problem between
densities that are supported on sets of unequal dimension. Recent work by
McCann and Pass reformulates this problem into a non-local Monge-Amp\`ere type
equation. We provide a new level set framework for interpreting this non-linear
PDE. We also propose a novel discretisation that combines carefully constructed
monotone finite difference schemes with a variable-support discrete version of
the Dirac delta function. The resulting method is consistent and monotone.
These new techniques are described and implemented in the setting of 1D to 2D
transport, but can easily be generalised to higher dimensions. Several
challenging computational tests validate the new numerical method