Since networking is disabled on emulators, it is not possible.
Desmume website states that wifi is not emulated and not supported. I assume the case is the same with the other emulator.
Another source (thanks @Robotnik)
From DeSmuMe FAQ:
Read more information about the current state of DeSmuME's Wi-Fi support. Right now, it's an experimental feature, which means it is unsupported.
If you are an end-user, then we strongly discourage the use of Wi-Fi. However, as an end-user, if you really want to use Wi-Fi that badly, then you are on your own. In that case, any posts ...
You can easily transfer the save files between different OSs. But the things that you need to keep in mind while transferring save files are :-
File name of the ROM (should be unchanged)
No$gba stores its save files as .sav file in the battery folder of the emulator.
Desmume stores its save files in the battery folder ...
Are you trying to trade between two instances of emulator or emulator and the handheld?
You can try https://www.pokecheck.org/ or if you google around, it's possible to find a downloadable GTS emulator (it's called fakeGTS), but I'm not sure if it's allowed to link to it here.