I don't know much about NTD, but neons aren't a very hardy fish. When you moved the tank, did the gravel change, etc., did you have the filter unplugged for a long period of time (more than a few hours) or did you change the filter / filter media?
My guess is that if the filter media dried out or was deprived of oxygen you're going through a new cycle and your ammonia/nitrite levels are pretty hight.
If you kept the filter running and media intact, it's probably just a case of adding too many fish at once. Adding one or two fish at a time gives the filter (beneficial bacteria) time to adjust to the additional bioload without having a significant ammonia spike.