They should do, but not always. The two main variables here are the type of fitting that you are using and the bulb itself.
Downlight fittings come in all shapes and sizes but recent regulations dictate that new build homes be fitted with fire rated downlights, which enclose the bulb in a metal container to prevent flames easily spreading through a ceiling in the event of a fire. The depth of the metal container used in fire rated downlights varies between manufacturers and obviously limits the length of the bulb that can be used. Some have plenty of space to accept GU10 LED bulbs that are larger than halogens, as many are, but many downlights are designed with traditional bulbs in mind so a very limited. If you are unsure, remove the bulb retaining cap from one of your downlights and gently push the bulb up into the fitting. If the bulb cannot move any further back then you know that you are limited to the size of traditional bulbs. For non fire-rated downlights the length of the bulb is less of an issue as they are not enclosed, allowing the bulb to push up into the ceiling cavity.
GU10 LED bulbs vary in size also, and with good reason. Ensuring long term reliability of LED bulbs means incorporating sufficient heat management to keep the internal circuitry from overheating. The more powerful the LED, the more heat they generate and therefore the larger the heatsink that is needed to keep it cool. If you are opting for high power LED bulbs then they will most likely be larger than traditional halogens so make sure your downlighters are deep enough to accommodate them. Be very wary of buying high power LED GU10 bulbs that are not slightly oversized as the chances are they will not have enough heat management and could fail early through overheating.