It's not a tendency. It's a cluster of monasteries/convents, all of which claim a particular man, Elder Dionysios, as their Elder.
Having visited three of his monasteries in Greece (two women's and one men's monastery), I would like to mention that they are quite controversial.
There seems to be quite a personality cult around elder Dionysios, and his style of leadership is very centered on his person. He forbids his monastics from confessing to anyone else in his absence. If he is not near, he hears confession through the phone. One of of his monasteries is next door to a home for mentally disabled children. He has forbidden the nuns from volunteering there. Instead, he has them raise sheep and goats, whose meat he blessed them to eat (!).
He is in conflict with several diocesan bishops in Greece, there even was one case when he blessed one of his abbesses to sue a bishop in a civil court. It is not surprising that they went to the OCA and later ROCOR, since there is no way GOARCH would have let him start a monastery in the US. He is on good terms with the MP though (if I recall correctly, he has met with Pat. Kyrill), and the Greek bishops seem to be unhappy with his entertaining such relations behing the back of the Greek bishops.
By the way, many (most?) of his monks and nuns are not Greek, and amongs those who are Greek, several have grown up aborad. There are many converts - Americans, Germans, former Russian Jews who have become Orthodox, also some Orthodox from countries other than Greece, especially the former USSR.
I cannot really say what all this means for the current situation in the OCA, but (and this is just my personal opinion), two things see to be possible:
1) Someone from the Greeks complained about the presence of the Dionysians in the OCA and/or
2) the Dionysians were seen as (and quite possibly, rightly so, but I cannot know for sure) as an alternative power structure, with the potential to weaken the positin of the local bishops and the OCA's Holy Synod.