The acquisition of multiple resistance to drugs is often viewed as a slow process resulting from the incremental accumulation of resistance to single drugs. We challenge this view with the analysis of the evolution of drug resistance in a pathogen spreading in a heterogeneous host population where drug use varies periodically in space. We study both the transient evolution of drug resistance at the front of the epidemic and the long-term evolution of drug resistance far behind the epidemic front. We show that multiple drug resistant pathogens can drive epidemic spread, while the evolution of single resistance may only occur in a second phase, when the pathogen evolves local adaptation to single drugs. Indeed, a pathogen resistant to multiple drugs can outpace the spread of a coalition of pathogens resistant to single drugs when drug use varies frequently in space. This result is amplified in finite host populations because demographic stochasticty can lead to the local extinction of maladapted pathogens specialised to a single drug. Our work has important implications for the optimal management of multiple drugs against pathogens but also for the optimal deployment of host resistance in agriculture.