Ireland, Germany, France, Italy, the Netherlands and Belgium are all halting flights. The measures vary and are initially short-term but the French rules also affect Channel freight.
An EU meeting on Monday morning will discuss a more co-ordinated response.
The new variant has spread quickly in London and south-east England.
Prime Minister Boris Johnson on Saturday introduced a new tier four level of restrictions for those areas, scrapping a planned relaxation of rules over the Christmas period for millions of people.
Top health officials said that there was no evidence the new variant was more deadly, or would react differently to vaccines, but it was proving to be up to 70% more transmissible.
Health Secretary Matt Hancock said the new strain “was out of control. We have got to get it under control”, admitting that this was “an incredibly difficult end to frankly an awful year”.
Within hours of the UK announcement on Saturday, the Netherlands said it would ban all passenger flights from the UK until 1 January.
Later on Sunday it said it would also bar ferry passengers arriving from the UK, although freight would continue.
The country on Sunday reported a daily increase of more than 13,000 cases – a new record, despite tough lockdown measures being applied on 14 December.
As Sunday wore on, major new restrictions were imposed by key European nations.
The new variant was first detected in September. In November it made up around a quarter of cases in London. This reached nearly two-thirds of cases in mid-December.
This variant is unusually highly mutated. The most likely explanation is it emerged in a patient with a weakened immune system that was unable to beat the virus.
There is no evidence yet to suggest the variant makes the infection more deadly, and at least for now the developed vaccines will almost certainly work against it.
However, if the virus changes so it dodges the full effect of the vaccine, then “vaccine escape” happens, and this may be the most concerning element.