Step 1 is dump the saves.
If using a flash cart it should be a matter of grab SD card (or plug in cable if it is one of those ones), copy and paste.
If using an original cart then there are loads of homebrew programs to dump saves. Stuff for the original DS will tend to revolve around either the GBA slot (either running from or copying to), wifi or the like and involve swapping out the DS cart if it is not being dumped from a GBA slot flash cart (unlikely in this day and age).
The DSi and 3ds on the other hand should have options to dump from them if they are hacked and can run homebrew.
There are also hardware devices that will dump saves in various manners.
Flash carts, and also some of the dumping tools, will use other formats that are not raw saves.
Most of the time this involves sticking a bit of info on the end of the save to identify the game (DS games were responsible for their own save formats which meant most games did not include any easily identifiable info, newer consoles will tell them to put their save data in this format which usually includes a name of the game), trimming them (most DS games do not indicate the size of save so most dumpers/flash carts then went for the maximum*) or compressing them.
To that end we have conversion tools
http://shunyweb.info/convert.php
*the actual maximum is considerable and seen in that warioware DIY game and some music games (daigasso I believe was one) where you download extra songs or store your own creations. For most practical purposes though then 512K was it, something of a problem when 2 gig microSD cards were a considerable expense and many were still using 512 meg or 1 gig hence the compression.
Emulators may also have limited abilities to accept other save formats (their own tending to vary, even more so if you go fiddling with options). Emulators will tend to be more raw formats than others but there are exceptions. Anyway said other format options might well be in the file menu somewhere to import a save, as opposed to simply naming it the same as the ROM and putting it in the save folder.
Also be sure you are using the same ROM -- while I would expect most Euro and NA ROMs to play together, indeed it is far from unusual for Japanese and Euro/NA stuff to play nicely, there can be differences, and likewise there may be differences for v1.1 or those games that got a later version with wifi support.