>>> On what grounds do they "face arrest" if they try to move from one Schengen country to another?
Unless a Brit has proof of residency within the EU, he/she can only remain there for a maximum of 90 days without a relevant visa. If, say, such a person tries to leave the EU (e.g. to visit family in the UK) his/her passport will be checked to look for an entry stamp, so that the length of their stay can be assessed. As there will be no such entry stamp for someone who has been living in Portugal then, unless they've got proof of residency, they potentially face arrest for overstaying within the EU (as it will be assumed that they entered before UK passports were subject to stamping).
While there will be no such passport checks if, say, such a person simply crosses a land border between Portugal and Spain, they might still face them (even when wholly within the Schengen area) when travelling by air.