We extend flux ratio theorems concerning ratios of unidirectional flux transients passed (in complementary experiments) through a medium of spatially inhomogeneous transport properties pertaining to diffusion, migration and temporary trapping of the transported substance. Any nonlinearity in the transport equations leads to a breakdown of the Ussing flux ratio theorem pertaining to all times. An integrated flux ratio theorem is proved for the case when the nonlinearity is in the kinetics of trapping, as when trapping sites can be saturated. The new theorem is shown to fail when the nonlinearity is due to a concentration-dependence of the diffusion coefficient, as in facilitated transport. The nature of a nonlinearity in membrane transport can therefore be elucidated experimentally by the use of the integrated flux ratio.