Derek and the Dominos started off with Brits Eric Clapton and Dave Mason as well as Americans Bobby Whitlock, Carl Radle and Jim Gordon. Their most famous work happened with Duane Allman in place of Dave Mason.
There are two bands called Hold Your Horses, a Swedish band with one American (does not qualify) and a French band with a couple of Americans, but I can’t find precisely how many. They have an awesome video for their song “70 Million” which has lots of visual gags art students would enjoy.
I’d be shocked if Pink Martini didn’t qualify, but all the members I can get information about are US citizens, though one guy was apparently born somewhere in Asia.
Does Up With People qualify as a “band”?
Can you clarify, though, if something like 2 Americans, 1 Swede, 1 German (just making it up) qualifies, or does it have to be something like 2 Americans, 2 Swedes to satisfy your OP? Or three or four members, each from a different country? I know you did say at least two members each from two countries, but I’m not sure if you actually meant that or not. It seems kind of arbitrary that 2 English 2 Americans would count, but not 2 English 1 American 1 Canadian, but it’s your thread (and half the respondents seem to be completely ignoring the premise as initially stated, never mind the later qualification.)
OK, how about Dio, then? That’s a little bit of a rough one and we might need to finagle a bit, as they’ve had a few lineups, some with everyone from a different country, it seems.
But let’s go with the line-up on Lock Up The Wolves. We have Dio, American, bassist Teddy Cook, American. Guitarist Rowan Robertson and drummer Simon Wright, both English. And then Jens Johansson, the Swedish keyboardist.
Jonathan Cain was born in, and grew up in, Chicago. As a grade schooler, he was a survivor of the infamous Our Lady of the Angels School fire.
Cain once played in a band called Bad English. I don’t think that counts. 
In the early 1970’s Gong had several British members (Tim Blake, Steve Hillage, Gilli Smyth) as well as several French members (Mirielle Bauer, Miquette Giraudy, Dider Malherbe, Benoit Morlen and Pierre Morlen) as well as Mike Howlett from Fiji and Daevid Allen from Australia.
Jefferson Starship had Pete Sears and Aynsley Dunbar from England, with the other members being American.
The Rising/Long Live Rock’n’Roll lineup featured American Tony Carey on keyboards as well as R.J. Dio on vocals.
The Joe Lyn Turner era always featured at least two Brits (Blackmore & Roger Glover) and three Americans (Turner, keyboardist David Rosenthal, drummers Bobby Rondinelli/Chuck Burgi.)
The 1987 touring lineup, arguably their most famous, consisted of David Coverdale, Vivian Campbell (both British), Adrian Vandenberg (Dutch), Rudy Sarzo and Tommy Aldridge (both American.)
During Rob Halford’s departure, Judas Priest consisted of three Brits plus two Americans: Tim “Ripper” Owens and Scott Travis, on vocals & drums respectively.
Paul McCartney & Wings had various lineups, some of which meet the criteria in the OP.
The basis of the group was always Sir Paul and Denny Laine (English) and Linda McCartney (American). Two of the drummers were American - Denny Seiwell and the ironically-named Joe English. So when one of them was drumming, there were at least 2 members from England and 2 from the USA.
I wrongly thought Jonathan Cain was English because he was the keyboard player for the English band The Babys.
Perhaps the Babys now qualify.
Swiss keyboard player Patrick Moraz was a member of otherwise English bands Refugee, Yes and the Moody Blues.
The later Babys lineup (1978-1982), in fact, does: Jonathan Cain and Ricky Phillips were American, while John Waite, Tony Brock, and Wally Stocker were English.
Does Wales count as a different country than England? If so, then Motörhead’s final lineup consisted of three members from different countries, England, Wales and Sweden.
Yes, at times they were more English than Australian. With the Young brothers and their nephew Stevie being born in Scotland, there is currently no Australian born member in the band. In fact original drummer Phil Rudd and original bassist Mark Evans were the only two members from any of the well known lineups who were born in Australia.
And they were half American on the Seventh Star album with Eric Singer and Dave Spitz, and then Australian Bob Daisley replaced Spitz on bass for The Eternal Idol.
The Afro Celt Sound Systemhas 2 from the UK, 2 from Ireland, and 7 from other countries.
Of course Crowded House had Kiwis Neil and Tim Finn as their songwriters, but their original lineup also included Aussies Paul Hester and Nick Seymour. And later they included Americans Mark Hart and Matt Sherrod.
Two guys from each of three countries…definitely a crowded house.
The Waco Brothers is another one, formed by Jon Langford of the Mekons. They’ve been through a couple line ups, but the early one had Mekons drummer Steve Goulding (so both Mekons from the UK) and Americans Mark Durante on steel guitar and Dean Schlabowske on guitar. Seems like the current line-up works, too, with UK-born Alan Doughty on bass and American Joe Camarillo on drums. Oh, and I guess a third Brit, Tracey Dear, on mandolin.
These are the former members of Deee-lite:
Supa DJ Dmitry aka DJ Dmitry (Dmitry Brill, Kiev, USSR)
Lady Miss Kier (Kierin M. Kirby, Youngstown, Ohio, United States)
Towa Tei (Dong-hwa Chung, originally from Tokyo, Japan)
DJ Ani (Ani Q. Scheme, Kansas City, Kansas, United States)
Deee-Groovy.