Ireland, Germany, France, Italy, the Netherlands and Belgium are all halting flights and travel. The measures vary and are initially generally short-term.
An EU meeting will be held on Monday morning to 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.