We study the (signed) flow of spectral multiplicity for a family of magnetic Schrodinger operators in R^2,
H(\lambda a) = (-i \nabla -\lambda a)^2 +V(x),
\lambda \ge 0,
in the large coupling limit, i.e., $\lambda \to 0$. Our main assumption is for the magnetic field
$ B curl \lambda a$ to have compact support consisting of a finite number of components. The total magnetic flux may be non-zero.