There aren't that many explanations for why your XL would become colder at night. The most apparent would be night-time room temperature and humidity, exposure to open air--other than if you'd just left it somewhere cold. In other words, there are no concrete reasons for why it would become cold internally (on its own). However, the DSi XL may be more likely than the DSi to get cold due to more physical space between components.
The lithium-ion battery, though notably efficient, does not react favorably to heat, for one "extreme". And batteries in general operate best when at optimal, normal temperatures, rather than starting up cold (i.e. car batteries).
Because of their fairly sensitive liquid-like film, the LCD screens should avoid being exposed to excessive cold or heat, so that might also explain why they didn't respond immediately. It's probably more so the case thanks to the XL's much larger screens.
If by "going slow" you mean a decreased frame rate in a game, that is rare. But could happen if affected by a chilled battery.
Game devices like these are designed to be resistant to such effects, but do usually bounce back after a few minutes.
My advice is simply to keep your XL somewhere where you know the surface and air temperatures will be moderate, and do not attempt to operate the system when you can tell that it is either slightly noticeably hot or cold.