Is this the same combination of DS card and GBA Cartridge? or are you trying different combinations?
Example: You are trying Diamond with Ruby. Have you got any alternative GBA or DS cartridges to test? Diamond and [any other GBA game] or [Any other DS game] and Ruby. As if a different game works, it's a cartridge issue; however if ANY combination won't work, it's a hardware or contact issue.
Is the DS detecting the GBA game in the slot when you start the console? as that's a key element, it needs to know it's there on startup. Can you play the GBA game on the DS at all? Can you save the game? Is the Clock running on the GBA game (potentially berry glitch or battery)?
If it reads the GBA game and can play it, i can't see any reason why it can't transfer to the DS - as it can obviously read and write from both slots if you can launch, load from, and save to both.
If a single cartridge combo won't work, there's the issue - check the GBA cartrdge contacts. If no combos work, or you can't launch/read from/write to any GBA game - then there's your issue, your DS's GBA reader is faulty. Check the contacts inside the DS then consider that the port itself may be dead.