I personally do not think that there are call variations in avian species across countries. Having been in a number of protected areas and other places within southern Africa, which share a lot of bird species in common, I find identical calls among individuals of the same species. In fact, same recorded identification calls (audios) are used across different countries, wherever such avian species occur. Perhaps, speciation and evolution theories can help explain this phenomenon.
I partially agree with vincent, however the variations exists due to a different factors encomprissed under different ecological drivers such as environmental (weather), clinal and also human disturbances (both are considered extrinsic factors).
Furthermore, this changes are not due to country border existence (if not geographical barriers exist) but also to other factors linked to behavioral changes at both population and individual levels (check this paper out https://scholar.google.es/scholar?start=30&q=intraspecific+variation+bird+call&hl=es&as_sdt=0,5&as_ylo=2015#d=gs_qabs&p=&u=%23p%3D_EfgL0FI-bkJ ).
In conclusions, it is feasible that those variations exista at different spatial scales, the challenge would be to unravell a set of variables to evaluate how those changes are linked to determinate factors and how they interact at both individual and species level.
Some good papers talk about this and other related issues, search them somewhere.
The calls and songs in birds are species specific and can be recognized by the members of the same species. It is not true that it differs between countries.
They exists intraspecific variations due to the development of dialects and subdialects among metapopulations. The thing is not the country but the isolation among population (e.g. when population was divided in isolated patches, thus in process of alreadu created metapopulations).
Question must be pose again with the right focus. Countries and boundaries are abstract sociopolitical concepts, without ecological basis.
It would be interesting to differentiate between bird calls (more short, simple with one specific context of alarm, group moving cue, etc.) and bird songs (usually longer and more complex, with territorial and mate attraction functions).
Though sharing some similarities that allows species identification, intraspecific variations in calls are common and are often used as flock or individual recognition (see the crazy example of penguin colony chicks) or may arise from the divergence between population as a consequence of a learning process (see the Grosbeak's example)
Article Finding a parent in a King Penguin colony: the acoustic syst...
Overall, geographical variation was more widely studied in bird song, and distant populations often show differences in full song repertoire (as in my nightingale model Luscinia luscinia, among many) or in a particular structural element (see this awesome dialect map from my colleague of the "Yellowhammer Project")
Article Repertoires and Geographical Variation in Bird Song
Fortunately, birds do not recognise political borders, so the same bird calls and sings the same in the US and just across the border line in Canada.
However, there is substantial research evidence that demonstrates the variability in avian acoustics across a region with different environmental, physical and ecological conditions....sometimes separated purely by distance, often by natural barriers, rarely by artificial barriers as well.
One bird could have several calls up to 1000 calls in some species. The calls of some birds vary with seasons. At certain time of the year, It may use some of its calls and at some time in the year it may not even call. Birds call may not change from country to country or region to region but may vary according to the season.