My mother has given me her caravan that she can no longer use. Is it possible to collect and bring over usi g my French registered car? What are the alternatives or way round the issue 
I don’t think so Debbie, and welcome btw, I can’t offer evidence but I do remember it coming up in the past and I offered the example of artics in Europe, and Britain, towing trailers of many different nationality registrations. It was then explained to me that the law allowed those specifically and not private vehicles including cars. No doubt someone else will be along soon to confirm or contradict that. 
Unless it is a very small caravan, like both of mine, (no more than 500 kgs weight) which doesn’t count as a separate vehicle in France and thus may be OK.
Alternatives? Hmm.
Take a French registered, or permitted, trailer from here and bring the caravan back on it?
Find a friend with a UK reg car to fetch it for you?
Pay for a commercial vehicle transport firm to deliver it, there are some in UK who specialise in collecting accident damaged cars from around Europe for insurance companies and maybe not have outward bound loads?
Sell it in UK and buy another one in France?
Unless the caravan is under 750kg (unlikely) you cannot tow it in France with a French registered car.
If it’s over 750kg you will not be able to register in France without obtaining a CoC and converting gas and electrics.
It will need to be declared for duty on entry.
The reason I think it is 500kg (unladen weight) and not 750kg is because that is, or at least was, the weight at which the trailer/caravan, needed to be registered.
Things may have changed since I was advised that by the DRIRE of course.
Just re-read your post and am wondering why you think it unlikely that a caravan would be under
750 kgs.
I have owned and used no less than 4 in that bracket since I came here to live. 3 British built and one German.
Yes, you are correct registration is required at 600kg. A mental lapse on my part conflating it with brakes required at 750kg.
Whilst there are a few caravans under 750kg it’s a minor part of the UK market.
No, it needs to be registered if it’s over 500kg GVW or PTAC.
That’s what I was told, never had to register one yet. 
I did extensive research as I brought a folding camper from the UK, with no clue as to how difficult it would be.
Surely not more than 500 kgs? So why was it difficult?
The specification says it’s 620kg ex-works and it was plated at 750kg.
I had no idea those campers were that heavy.
That’s just about the smallest recent model. We used to have a Pennine Fiesta from the 90s which was smaller but even that was 460kg with a GVW of 650kg.
It’s really frustrating as I really liked my folding camper and, to my knowledge, it’s very much a UK thing and there isn’t one with proper EU certificate of conformity.
No it isnt. The Dutch and the Danes use them far more than the UK. Many have been ditched over the last few years for proper caravans probably due to rot and old age.
I’ve been googling for years and have never seen a folding camper in the same style as Conway Countryman from a mainland European manufacturer that can be registered in France, are we talking about the same thing?
If there’s something I’ve missed than I’d be really pleased. Maybe a different search term?
Edited to add: I don’t think I’ve even seen one with a non-UK plate after many years of European camping with one.
I used to live on the Dutch - German border and saw lots of them coming across mostly dutch registered. No idea what the manufacturers were as I am not a “van” person.
I suspect we’re taking about different things as I’m talking about one of these. When folded down it looks a lot like a trailer tent, rather than one if those elevating roof caravans.
That is what I was referring to. not the elevating type.
Raclet, Cabanon and Trigano manufacture them here in France
No they don’t, not like that. They make trailer tents which are completely different.
That is what you pictured A TRAILER TENT. Exactly what the dutch and the Danes use/d