There have been some recent cases of British MEPs being suspended from their party "at home". What are the implications of such moves for their activities (and EU-level groups) as a MEP? Do they lose their membership in the corresponding EU Parliament group, or is any of that subject to subsequent voting that involves more than their national party (group)?
MEPs who are suspended from their party would remain MEPs. They may or may not remain part of the group in the European Parliament, but in the cases I have seen they would also leave the group.
Richard Ashcroft, was suspended by the Tory party in Oct 2017. He remained in the ECR group until Feb 2018. He then left the ECR group and joined the EPP group. He was free to do this as he was not subject to Conservative party rules (that require all their members to join the ECR group)
Janice Atkinson was expelled from UKIP. She left EFDD and joined EFF.