to think the Idiot is wrong?

The children have met the 'new' (or not so new) girlfriend twice before at his new home without DD's knowledge.

The Idiot is convinced she is jealous but DD actually considers herself well off out of it!

He is having the children today as he is not working and, when she brought up the subject of needing some sort of a routine for the children and for them to be able to stay over, he picked her up by the lapels and threw her out!!

Apart from the fact of it being confusing for the children I feel it would be an appalling example to set if he went from one new gf to another.

I agree that there should be a gradual introduction to parents new partners and this should be done outide the home. For example taking children on an outing and asking if they would mind if "new friend" came along too. It would be awful for the children to visit the idiot their father and find another woman (not their mum) staying there. I agree with your DD kittylester his "new" girlfriend should stay away for a while and gradually meet the children on neutral ground.

Once this "relationship" is established and serious, though I can't see a problem and awful as it may seem DD would simply have to go along with it as she doesn't want to have to make her ex choose between his children and his new partner.

Are the children familiar with the new woman? Does she spend time with them on the days the idiot has them?

No, I think they should have a gradual introduction. I sent DD back to stay with her father and she was horrified to find a strange woman living in our old house. I was equally horrified when I went to collect DD to find her putting up xmas decorations - the relationship had been going on for ages before I left him and I knew nothing of her existence! So later on when I met DBH I waited for weeks before he was allowed to stay the night. Months even. I wanted her to be comfortable in her own house with him and check for myself that the relationship would be ok.