A magnet is in essence made up of atoms with orbital and spin angular momentum, (mainly due to the electrons) and the forces that act on these electrons can be derived from Dirac's equation, but give you what is essentially the Lorentz force law. You can think of the magnetic field acting on every single electron individually, and all these forces added up will apply a net force and a net torque to the magnet as a rigid body.
If you want to do some calculations, you can imagine each atom as having a microscopic current proportional to the dipole moment of that atom (basically, the magnetization). This "bound current" is
$\vec{J} = \nabla \times \vec M$.
Again, imagine an electron orbiting the nucleus as providing a "current" flowing around the nucleus -- the Bohr magneton. It's just an idealization but you can make it rigorous if you want. Now, currents of adjacent atoms cancel out where they intersect, because they are going in opposite directions, but on the boundary of the magnet they do not cancel out, because there is no neighboring atom there. Thus it is perfectly mathematically and physically sensible to model a permanent magnet as a sheet of current moving along the magnet's surface, roughly like a solenoid does, and given by
$\vec{K} = \vec{n}\times \vec{M}$,
where $\vec{n}$ is the normal to the surface. Now this surface current is a real current, and therefore it will experience a force when subjected to an external magnetic field (and it's own magnetic field, btw. This is why transformers hum).
