Consider you have an Ising lattice with a dominant up component out of thermal equilibrium, that's your initial state. The down spins want to flip up and align with the ups, and they'll do so until a thermal distribution is reached. While they flip, the overall magnetization increases. Does that create a current by induction? I haven't been able to find a good reference.
According to Maxwell's equations, a changing magnetic field creates an electric field, not a current.
This effect would indeed happen, but the Ising model just doesn't contain it. It also assumes that you have "spins on a lattice", which also don't really exist in this form (in reality, they are electrons which belong to atoms in a crystal lattice), and there is a bucketload of other effects which are also not considered.