Not being rude here and perhaps most people don't share this view but you come across as rather unreasonable. New people moved in and it sounds like the first interaction with them was to passive aggressively ask them "are you still moving furniture around". I'm pretty sure you knew they weren't because little feet running around don't sound anything like furniture being moved around or adults moving about. The rest of your post reads somewhat like someone who is looking for issues to try and get them kicked out....because of the noise rather than for any of the other issues.
You live in a flat. Some flats have good sound proofing, others don't. They have young kids. Young kids have the attention span of a gnat and don't understand that they can't run about all the time because its noisy for others. You can tell them this until you are blue in the face and they will still do it. How old are the kids? Are they 2? 5? older?
As a parent you can try to encourage different sorts of play and behaviour but at a young age there are limits and those limits are not very big. If they have twins then its worse. I have twins and they wind each other up and want to play together. It would be a nightmare to try and stop them from making noise.
I'm not sure you are going to get any joy with noise complaints from children running about above you. Its not antisocial and its not something that can be easily remedied. You might simply have to look to move or look to put sound insulation on your ceiling to help with it.
It sucks but in any accommodation where you share walls/ceilings etc with others you might have noisy neighbours. We had neighbours in our last house who had 2 spaniels and three cats. The spaniel would nearly give me a heart attack regularly because he would wait quietly until I was walking past the fence and then bark at me like a nutter. One of their cats sat in our front garden 80% of the time and people would just walk up to our house to pet the cat. Another one was nasty and would hiss at and attack our cat in our garden and the third came in and ate our cats food whenever he could. They were great neighbours and in the grand scheme of things, these were minor issues but we could easily have fallen out with them about it.
Constant noise is horrible so I understand that it makes you a bit irrational but you need to try and work to resolve this rather than just making enemies. Can they put rugs or carpet down if they have cheap ****** wood floors. Can you sound proof your ceiling. If your neighbours like you they are far more willing to try and keep noise down rather than just thinking "**** him".