Of course today, anyone should be able to pursue anything they want to pursue. When anyone expressed the interest in studying a specific art, I would try my best to direct them to what they desired. Not that my students ever had that need.
But when you look at the concept behind cross-training, there are interesting things to consider.
If you mean acquiring skilled training in another art, besides the one you are pursuing, that is not an easy thing to do. Your new art really is not someplace to continue training in your prior art. That would have to be done on your own time, not theirs.
And what do you consider cross-training? Putting in some time to gather a few techniques is not really spending decades with a qualified instructor under constant guidance to acquire that art. That does make a difference for receiving a few techniques does not constitute being trained in an art in my book..
When I was a new black belt I was without any guidance in my area.
I took up friends offers at tournaments to visit and occasionally train with them. The reality was I was only looking for a place to find someone to spar with. As it turned out at none of those places I trained, did they ever have sparring while I was there.
I did take the advantage to work out with other martial artists, but my intention was never to try and learn their art. I was pleased that I had my own art to work on, but anything I saw I learned to remember, often taking notes on what was shown. So I did acquire something, but nothing like their art either.
I did specifically pursue one art. In college on my studies of Taoism I learned about Tai Chi Chaun a bit. I was interested and when the chance arose I asked Ernest Rothrock to be allowed to study it with him. He agreed and in the course of 2 years of ½ a week lessons I acquired the form, and a bit more. Then over the subsequent decades he provided some further guidance. I really am not a trained student of his, just a bit trained. And I did learn a lot in that process.
Almost at the same time, I did have a lot of free time.. between training, teaching, visiting others and the tai chi, that took much of that free time. But one person I met on the tournament floor was Tristan Sutrisno. I took advantage of his offer to visit. I saw so much complete different from what I knew, and in his way he tried to test me, discovering I remembered much of what I was shown.
I learned many things. Knowing someone from a tournament, watching them compete, watching them judge, observing their students, meant you knew nothing really about their art. At that time he enjoyed my learning a bit of his system.
A little later I approached Ernest with a different request. I did not want to be a student of his arts, I respected what that would mean too much. But I realized I often had to judge Chinese stylists at tournaments, and realized that I really did not know what I was judging.
I asked him if it would be possible to learn some Chinese forms so I might become a more informed judge. Now he had been training me for 9 months at that time. I think he thought it would be interesting to see what a karate guy could actually learn.
His studies had covered many arts, and he know a Herculean number of forms. He elected to teach me advanced forms from systems his students were not studying. Perhaps as an experiment. In any case I learned a few forms (around 20 or so).
None of this made me a qualified student of these instructors. They all knew where I was trying to train. They shoved and shoved more into me.
Then the time came when I had to move, again I was on my own. Working hard to work on what others had shared with me. But now my other responsibilities grew. Personally I had to make very hard choices what to retain, and I did.
My youth program grew and now I had a small adult program, I elected to add some of those studies I found valuable as subsidiary studies for my students. Of course I could only share what I understood, and more that a little of flavor of Isshinryu attached to them.
I have too much respect for those who shared with me to ever consider myself trained in their at. And most of them have been able to share with my students too, to let them see from where those drills or kata came from. Their sharing has most definitely shaped my art. And I was proud that I was able to share some of that training with my students.
Of course this was way before I ever heard of cross training.