I have many issues with the mother being "free" because she becomes uncontactable, doesn't show up on time for the kids etc.
why should she be contactable? the children are with their father, their other parent, where they are (hopefully) safe. Why would she need to be contacted? In the event of the unforseen, a message can be left on her mobile and/or house phone and/or with friends and/or on Facebook etc. etc. Presumably, as a parent, your partner is capable to dealing with whatever issues might present themselves when he is with his children? She will need to deal with the issue if she doesn't respond to urgent messages (and accept the consequences of that, if necessary) but as someone who values her time away from her children, I can honestly say that in nearly 7 years, there has never been a reason for my ex to need to speak to me during his contact time.
If she is not turning up after contact, what time frames are we talking about? 10 minutes, 1 hour, 10 hours? If it's game playing and being difficult for the sake of it, often just ignoring it and saying nothing will change the behaviour (although possibly to something even worse) because it's not having the desired affect (which is upsetting the ex).
Of her five children, all of whom have different Dads (I'm not judging her for this), he is the only one which has stuck around
if you're not judging her, why mention it? why is it important that we know that? your partner, after all, considered her a reasonable person to be having children with so if you want to judge her, judge him as well!
Out of interest, who moved? Your partner or the ex?
I am a little troubled by the idea that some people are suggesting that your partner should be encouraged to use a hotel rather than keep a home. Much of it will, of course, depend on affordability and his contribution to your household, but having a home for the children is very different to seeing them in a hotel.