A lot of respondents aren’t answering the OP as intended: 2 or more members, from 2 or more countries.
Transatlantic fits, on a technicality. The band officially consists of two Americans, a Brit, and a Swede. But when they tour, they have a fifth guy helping out on guitar, who is also a Swede.
Buffalo Springfield was three Canadians and two Americans–or the other way around after Jim Messina replaced Bruce Palmer.
A lot of salsa bands have both native-born Americans and native-born Puerto Ricans, although of course Puerto Ricans have US citizenship.
The Joe Cuba Sextet, who had a crossover hit in 1966 with “Bang Bang,” were a Latin group in this category.
Fleetwood Mac wanted to just add Buckingham but he told them that he and Nicks were a “package deal” since she was his girlfriend at the time.
And when Buckingham/Nicks left in the 80s, they were replaced by 2 Americans Rick Vito and Billy Burnette
The Police were two Englishmen and an American.
The Velvet Underground had a Welshman and several Americans.
I was gonna say…! 
Trevor Rabin was a South African in an otherwise English Yes lineup.
The Faces briefly had a Japanese bassist, Tetsu Yamauchi.
Ritchie Blackmore’s Rainbow had Americans Ronnie James Dio and Joe Lynn Turner on vocals.
The British band UFO had its greatest success with German guitarist Michael Schenker.
Not at the same time though, and we still need two or more from each country.
Journey’s original drummer was a Scotsman, Aynsley Dunbar. And their longtime pianist Jonathan Cain is English. And their current singer is a Filipino.
I was right when I said it was easy to find bands with 1 person from a different country than the rest of the band. 
Too bad those examples don’t fit the criteria.
It looks like Rhapsody of Fire is currently 4 Italians and 1 German, but last year’s lineup had 2 Germans. Until 2013, they also had an American.
Whitesnake has had multiple Brits and American’s in their lineup. I’m not going back trough each permutation but I’m guessing at some time they had more than 1 of each.
**Pretenders **= 1 US and 3 Brits
Vampire Weekend = 1 Sri Lankan (Rostam Batmanglij) and 3 US
The Police = 2 UK and 1 US (although Stewart Copeland mostly grew up in the Middle East since his dad was a CIA regional director)
The 80’s touring version of Genesis had three Brits (Collins, Banks, and Rutherford) and two Americans (Chester Thompson and Daryl Steurmer).
While ABBA is thought of as a Swedish group, only Agnetha, Bjorn and Benny are Swedish. Annt-Frid (or Frida) is the product of a World War 2 liason between a Norwegian woman and a German soldier.
The Dave Matthews Band kind of qualifies. The current line-up has four mainland Americans, two foreign-born (South Africa and Germany) and a guy from the US Virgin Islands.