Our French car is registered with them and is foc, when I get a new Blue Permit later this year if I pay £10 I will be exempt from congestion charge in Central London for the life of the Blue Permit.
Yes, they do chase you back to France, or they pass the details to a commercial company in France/Germany etc., who do the chasing for the TFL and their exorbitant fee. There was a report a while ago of one hapless French driver who ended up with a bill for nigh on 2k€ after not realising he should have paid for the Thames crossing, he went off on a longish trip, claims he was not at home so did not receive bill from company which doubled virtually every week.
He refused to pay, action was taken in French Court which was thrown out on some technicality but if he tries to get back into the U.K. he will be arrested and the vehicle impounded allegedly. Time to change vehicle and name methinks….