I have a question about children, how they learn and whether so called manipulating behaviours and defiance is learnt from watching other children?

I am not judging here, I want to figure out the dynamic and if I can find common ground, reassure my family.

In essence I live temporarily with my family, and next door we have person A, B and their children F3 and Fbaby. Mum A has a lot of fear about my children being in contact with her children, because she thinks her children learn how to behave "badly". She has not actually spent much time talking to my children, or me, or even spent much time with them at all apart from last year after we had a big life change and my children were actually quite traumatised. Their dad has arranged 4 international moves, isolating us many times, and I returned to family because I was distraught and needed some time to assess options.

The only evidence she has to go on that my children are infecting her children with "disobedience" is that she says her 3 yo is less "obedient" after spending time with us.

I have noticed that her daughter (3) wanted to come and see us, and since we are in the next house with my parents, she walks over to see us. We did not realise at first there was an issue, because we are all family and F3 has done this often before. But her mum wants her to ask for permission because my family are here and I am authoritative/liberal rather than highly authoritarian like her. I said to N that when she wanted to come over to tell her mum that she wanted to, and her mum said no not tell, ASK. Their children's emotional life is ignored, and I see threats, bribes, punishment,name calling as a go to.

I am wondering if their daughter, seeing me listening to my children and responding, has made her realise her emotional life is okay or that it exists? I noticed that she will do things my children have never really done, like make things up to get attention, i.e. telling me one of my children hit her when I had actually seen that all that happened was they were playing and my daughters said we are going inside now, bye, and ran in. Or she will pretend to cry, or use coy facial expressions and baby talk. She also will not let people touch her, or cuddle her, she pushes away physical touch which is something I have never seen before? My parents have looked after her for 1-2 days a week since she was a baby and she rarely touches them or cuddles. After a few days of knowing me, she was often cuddling me and clinging to my leg, and sitting on my knee, which I think threatened her mum as she is not allowed too much touch as it is seen as spoiling.

When I see those behaviours, I try to see what her need is, does she feel she is not heard, so I listen, does she need attention, does she need to rid of tension, so I make her laugh. It seems that mum A is dismissive avoidant, and dad B the same perhaps fearful avoidant. (If you know attachment theory I am more preoccupied!). And I think we are polar extremes so find it hard to understand eachother.

Would you have any ideas on how to reassure them? I am moving out as soon as I can, in the next few days, but it would be preferable that the relatinship was restored.