Yes, they fit and play perfectly. The DSi / DSi XL cartridge slot is **physically identical** to the DS Lite / DS Phat slot. It accepts all standard Nintendo DS game cards (NTR-005).
DSi / 3DS are **region-locked for DSi/3DS software**. Standard DS games (NTR-XXX) are **region-free** on all hardware.
Bottom Line: Your normal DS games work fine in a DSi. You lose GBA backward compatibility only.