It sounds to me like the DS is at fault. Either a wonky slot 1, or some other problem causing it to hang.
The only other possibilities I can think of are that the card reader you're using is somehow causing this, or the roms are corrupted in some subtle way that causes them to hang, but still allows them to start in the first place. Both of those seem unlikely given the circumstances, but stranger things have happened.
If possible, try using a retail DS game in the DS and see if it crashes, and try the R4s in a different DS and see if they crash it as well.
Edit: I just re-read the first post. The DS is powering off during gameplay? That's almost certainly a problem with the DS or the battery.