In my opinion, there's a difference between exploring other aikido styles through seminars or the occasional visit to a differnt dojo on the one hand and actually training in another dojo on the other. I have little understanding for dojos forbidding the former, but I do have some sympathy for dojos forbidding the latter. When you join a dojo, you're expected to conform to their style. (In due time, of course, you can't just flip a switch to switch styles.) Training in another dojo with a different style or regularly returning to your previous dojo, is not the best way to confirm your membership of the new dojo.
Actually, my own experience differs... Back in the early 80's I was relocated to the Seattle area. Saotome Sensei told me I should train with Mary Heiny Sensei whom he knew from Japan. So, I was a member of her dojo while it was still understood that I was Saotome Sensei's student.
Then Bruce Bookman Sensei moved to Seattle after training in Japan with Chiba Sensei. He did various things that Heiny Sensei did not do so I also paid dues at his dojo and split my time between the dojos. This, despite the fact that our teachers had a number of differences and didn't really get along. We simply decided that it was unnecessary to carry over issues that started years before in Japan that had absolutely nothing to do with us.
When Mary Heiny Sensei left for Canada in 1986, was asked to take over the dojo, which I did for three years. We were members of Chiba Sensei's Western Region of the USAF. I was given Fukusdhidoin papers, actually signed by Yamada Sensei, although I am sure Yamada did not know who I was and it was really done by Chiba Sensei. So, I trained at two different dojos while being a student of a third teacher, then ran one of the dojos in an organization run by a teacher, not my own.
I look at myself as a triumph for the idea of getting past political BS and just being able to train. In 1989 I opened Aikido Eastside which is within the ASU. I am still close friends with Bookman Sensei and Mary Heiny Sensei as well... Bookman Sensei recently asked me to participate in his 30 year anniversary seminar since I was actually a member of his very first dojo.
People should be free to train wherever they want. When it comes to Rankings, well, that's another story. If someone wants a rank from me, especially a Yudansha rank, they need to be training with me and supporting our dojo. Giving rank is really creating an association in people's minds between a certain student and a certain teacher. But if a student doesn't want or require that from me, he or she is free to train whenever and with whomever they wish. All my teachers knew that I would get whatever rank advancement was appropriate from my own teacher, Saotome Sensei.
While I understand that my own experience is almost totally unique, at least in the US, I think it points out that everyone can benefit from putting all the politics aside. My own training was better because of the breadth of teaching from very different teachers. I like to think I was an asset at each of the dojos I trained at, helping each to be a bit better because I was there. Anyway, this was only possible because some really wonderful teachers cared more about the art than the politics and a couple of Japanese Shihan chose not to impose their differences with each other on me. If things were more this way as rule, we'd all be a lot better off.