There was a whole era where it seemed every ice dance team had a Russian coach with the Italians and the French being the exceptions (at least with high profile teams). I think because after the 1980s, it was as if "that Russian style" was the only style that was being rewarded. Even Bourne-Kraatz, who people associate with Canada and being THE "non-Euro" team, had mostly Russian/Soviet-trained coaches during their senior career as medal contenders.
Even singles skaters from all around the world started to go to Russian ice dance coaches too or at least Russian ones even if they weren't single skaters themselves. Remember when it seemed all of the Japanese ladies moved to train abroad and Russian coaches were a big part of that? I think they went to mostly American and Russian coaches because they saw that was where the success was. Examples:
Yoshie Onda moving to Audrey Weisiger (who I felt gave her some great programs) and then moving to Josee Chouinard. She didn't get a Russian coach, but she was part of the wave of Japanese ladies going to foreign coaches during this period.
Fumie Suguri trained with Oleg Vasiliev for a while in 2004-05 but she was "ahead of her time" and the Japanese Fed didn't let her continue training with him, so she went back to Sato after Ando left him. However, she would go back to him after the 2005-06 season. Then in the 2007-08 season she went to Alexander Zhulin (who had choreographed her 06-07 programs. Apparently when she spent most of her time with Igor Pashkevich due to Zhulin being busy dealing with personal issues. Then she went to Morozov for the 08-09 season (where she qualified for Worlds for the first time since her silver medal winning 2006 appearance), then for the 09-10 season, she went to Mishin but apparently he "neglected" her and she was mostly learning from...Igor Pashkevich.
Arakawa moving to Callaghan and trained with him for like two years (I believe) and then moved to Tarasova/Platov right after the 2003-04 GPF when Sasha left Tarasova...then won Worlds but had an iffy 2004-05 season including a 9th place finish at 2005 Worlds, so moved to Morozov right before the 2006 Olympics after failing to qualify for the 05-06 GPF (but she listened to Tarasova who told her to go back to "Turandot" for Torino).
Ando left Sato and went to Carol Heiss-Jenkins having programs done by David Wilson...then getting an unfortunate haircut that aged her right before the Torino games, and then going to Morozov the next season and then...well, we know the storied history between Morozov and Ando.
Mao Asada moving to Arutunian but personal issues forcing her to go back to Japan and Rafael breaking off their coach-student relationship (which he now admits he handled incorrectly and is sorry about it since it had to do with insurance and issues of succession it sounds like after her mother's death) and then her going to Tarasova.
Even some of the men...Daisuke Takahashi was with Morosov (I don't know when he started but I remember his Morosovian programs as early as the 05-06 season) and then left after the 07-08 season.
Nobunari Oda was with Lee Barkell part-time and then moved to Morozov right when Daisuke left him, but went back to Barkell after the 2009-10 season.
But none for Yukari Nakano who stayed in Japan.
Even "America's Own" Michelle Kwan wasn't immune. After the 2002 Olympics, she started getting programs choreographed by Morozov (had Arutunian as a coach...not Russian but was probably trained under the Soviet system), and then after her worst result at Worlds in a decade (4th place!), she went to Tarasova to get her programs choreographed in preparation for the Torino games, which we know never happened.