The water is not what "cycles". It is a population of bacteria that live on your gravel (or sand), glass, decorations, and filter. Hence, changing the water doesn't slow it down because you remove practically 0 bacteria - you just take out the fishy poison.
You can remove 100% of the water in the tank, but as long as the filter and gravel stay pretty moist, your tank won't have to start over on its cycle. You are smart to save some when moving though, unless you know the water has similar GH and pH at the new place.
Do NOT listen to those telling you to hold off on water changes. They will keep your fish healthy during this stressful time, and not slow you up a bit as long as you have some ammonia and nitrites in the water.
Edit: I can post a detailed explanation of WHY changing the water doesn't slow down the cycle, but you should just trust me