This is becoming more and more popular. Many years ago i can accross the Mongol rally that's about taking a very much underpowered small car from London to Ulanbataar. That takes some cash and mostly time, so i never managed, even though my Suzuki Carry would be absolutely perfect for it. Cars can not be left there...
Then there is the Bamako rally, where you can join up pro rally teams with something hopelessly not suitable for that kind of job, and then you get to enter for free or something like that.
There's a Polish Zlombol that means communist cars, or cars designed during the communist era, so Skoda Favorit still plays... lot of Ladas, Yugo, Wartburg... That happens on road though, no abuse at all. Cars are supported through companies for advertisement on the cars. The money goes for charity.
there's a small Gruz rally think, organised entirely privately, where you enter with something not off-road capable and go to places which are considered off road. They went to Romania last year, and Felicias are very much welcome.
I guess as cars became very much available and still not too horribly expensive to run (I mean here in Eastern Europe) you can really get a still decent 30 year old car for a box of doughnuts, and have fun with it. That was unimaginable 10-20 years ago when a working car had a much much bigger value.
This year I planned to enter the gruz rally (Zlombol just became waay too big, last time they went with 5000 teams to Albania) with my not much loved Opel Astra Classic.
There's a guy who is running a Fiat Cinquecento with massively tractor like tires and goes everywhere with it. Cinquecento and Seicento were built here some 100km from me, and they are dirt cheap.
Even the Mongol Rally organisers (theadventurist.com i think) started to make more local rallies, for people who can't afford to drop life for 3-4 months... Now you can start and drive around Europe.
In that case Ricardo's comment is legit though.. you might end up scrapping your French car in Estonia, but let's treat the EU as one big car economy. After all I did the opposite, went with a wreck from Hungary to the UK, had an accident in Belgium, and eventually sold the carcass for metal in England.
I'm not sure about abuse or no abuse... the sheer fun and memories are worth for me, though I personally would do my best to bring the car back and keep it, complete with the stickers, scars, modifications. A Suzuki Carry that limped back from say Kyrgistan would be on a plinth in my garage!