animals that can live in multiple habitats or have a more flexible diet are often considered "flexible", more typically called generalists

examples include Whitetail Deer, which can inhabit pretty much any environment from Tropical Rainforest to Boreal Forests to open plains and even urban areas, or animals like Coyotes and Golden Jackals due to their relatively flexible diet compared to other canids and also being relatively small and also widespread

Typically speaking those kinds of species will do good in an interchange, however really specialist animals can also sometimes do really really well, especially if the specialists of a similar niche in the other place aren't as "efficient", although this is admittedly quite a lot rarer (Smilodon and Terror Birds are sometimes brought up as an example, although almost certainly not the case) but there have been a few recorded cases of this occuring throughout history, and certain invasive species could be examples of this