There is increasing evidence that global climate change will alter the spatiotemporal occurrences and abundances of many species at continental scales. This will have implications for efficient conservation of biodiversity. We investigate if the general public in Denmark are willing to pay for the preservation of birds potentially immigrating and establishing breeding populations due to climate change to the same extent that they are for native species populations currently breeding in Denmark, but potentially emigrating due to climate change. We find that Danish citizens are willing to pay much more for the conservation of birds currently native to Denmark, than for bird species moving into the country--even when they are informed about the potential range shifts associated with climate change. The only exception is when immigrating species populations are under pressure at European level. Furthermore, people believing climate change to be man-made and people more knowledgeable about birds tended to have higher WTP for conservation of native species, relative to other people, whereas their preferences for conserving immigrant species generally resembled those of other people. Conservation investments rely heavily on public funding and hence on public support. Our results suggest that cross-country coordination of conservation efforts under climate change will be challenging in terms of achieving an appropriate balance between cost-effectiveness in adaptation and the concerns of a general public who seem mostly worried about protecting currently-native species.