This is certainly frustrating and can be hard to deter but with some time and patience - it can be resolved.
Cats are crepuscular, which means they are instinctively most active during at dawn and at dusk - which are the best times to hunt. Domestic cats have the similar traits but their nightly timing is often slightly off. Cats with excessive energy, at any age, will also be prone to this behaviour and with multiple cats in the home they learn by way of observation learning to pick up each other's habits.
The first thing to do is ignore this behaviour. I know this seems impossible but it is the most important. By responding to them (at 3am or 4am or any unfavourable time) is providing the attention they demand and just reinforces the behaviour. This could be talking, cuddling, feeding or even angrily shouting.
So ignoring won't be easy and will likely take a couple of weeks. See how things go once you have made the big move. A new home will set a new routine and schedule and you may not have this issue anymore. However, should the behaviour continue try ignoring them at night. If after a few weeks...nothing has changed, you may need to confine them to a certain room away from yours - equipped with food/water bowls, beds, scratching posts and litterboxes.
Once in the new home, try providing their meal late in the evening (just before you retire for the night). If they are over their ideal weight - just provide a snack, such as a treat or a few dry kibbles. Have a play session of about 30 mins with wand toys to tire out the ones with excessive energy. Keep blinds and curtains closed during the night to block out the sights of wild animals or strays in the area. Bid them "good-night" with cuddles and pets and leave them for the night. Set a time, such as 8am or one that works for you and keep it consistent. Eventually they will associate this as the time you will tend to their needs.