A pathogen can readily mutate to infect new host types, but this does not guarantee successful establishment in the new habitat. What factors, then, dictate emergence success? One possibility is that the pathogen population cannot sustain itself on the new host type (i.e. host is a sink), but migration from a source population allows adaptive sustainability and eventual emergence by delivering beneficial mutations sampled from the source’s standing genetic variation. This idea is relevant regardless of whether the sink host is truly novel (host shift) or whether the sink is an existing or related, similar host population thriving under conditions unfavourable to pathogen persistence (range expansion). We predicted that sink adaptation should occur faster under range expansion than during a host shift owing to the effects of source genetic variation on pathogen adaptability in the sink. Under range expansion, source migration should benefit emergence in the sink because selection acting on source and sink populations is likely to be congruent. By contrast, during host shifts, source migration is likely to disrupt emergence in the sink owing to uncorrelated selection or performance tradeoffs across host types. We tested this hypothesis by evolving bacteriophage populations on novel host bacteria under sink conditions, while manipulating emergence via host shift versus range expansion. Controls examined sink adaptation when unevolved founding genotypes served as migrants. As predicted, adaptability was fastest under range expansion, and controls did not adapt. Large, similar and similarly timed increases in fitness were observed in the host-shift populations, despite declines in mean fitness of immigrants through time. These results suggest that source populations are the origin of mutations that drive adaptive emergence at the edge of a pathogen’s ecological or geographical range.