Certainly over here - you use one of the assigned DPE's for your specific region.
Which is probably the correct etiquette.
It's very common in the US for students to use designated examiners in or from another location. Particularly for multi-engine ratings, seaplane ratings, and ATP certificates and type ratings, students often travel to locations away from their home in order to get certification.
You're correct that in general, designated pilot examiners don't typically travel to get work, but then they don't have to. Most pilot examiners are too busy as it is.
Then again, the entire designated pilot examiner system is legalized extortion; taxes are paid to allow the FAA to operate, and individuals shouldn't have to pay designees to take a practical test. I find it a reprehensible program. When some students pay as high as six hundred dollars for a practical test, it's most certainly unethical, and borderline criminal.
These people have set up a business (many do have other occupations as well) and done whatever had to be done to get approved by the FAA.
If everybody who is qualified were to be given the same opportunity, that would be one thing. If the FAA regulated the DPE's and unified or standardized the fees that they charge (but shouldn't be able to charge), that would mean something.
I have no sympathy for the designated examiners, and they certainly have no place to whine if another DE steps up and flies with students.
Europeans really shouldn't have to go to the USA to get pilot certification, especially just for a piggy back certificate. For those who do make the trip, I hope it's a great experience, but there's no reason in the world that folks in the EU/UK shouldn't be able to do a checkride with a designated examiner closer to home.