Training with both Zen and Rimé I am very fond of the Prasangika Madhyamaka following Candrakirtis interpretation of Nagarjuna. This is what I read and study most. But I am also attracted to Zen. 

Does anyone have any thoughts, moral or otherwise the, on pros and cons if I want to train with teachers at both a Zen center and a Tibetan Rimé center?
The traditional view on this is negative. One explanation I heard, is that a particular teaching is like a raft made of simplifications and approximations - which has all kinds of holes and inconsistencies, is leaking enough already, and has no value on its own outside of getting you to the other side. Now, when you study multiple traditions, you're effectively building two slightly different rafts at the same time - in your head - which leads to a combinatorial explosion of contradictions. More leaking. You may never get to your destination. Am I making myself absolutely clear? :)

This is different from the Tri-yana approach practiced by modern Karma-Kagyu school (and pioneered by Rime's founder Jamgon Kongtrul the Great). In Tri-yana, you first practice self-discipline at "hinayana" level of simplistic but well-defined black-and-white concepts and rules, then graduate to "mahayana" level of going beyond dualities as you work with others, until you mature to the "vajrayana" level where you practice spontaneity, confidence and intuition. Each level is effectively almost a different school, but practiced in succession - not at the same time.

Now, in this particular case, the two traditions are Madhyamaka and Zen. Both come from Nagarjuna/Prajnaparamita movement, so both essentially speak about the same thing (Emptiness). The key difference is in the approach.

In Madyamaka you study and study and study the theory, until you know it all in and out, and then meditate, meditate, meditate for years - until one day hopefully it clicks and the theory connects with experience. Essentially, Madhyamaka is walking back and forth on the hither shore while talking about the other shore - until one day you get used to the idea of the other shore enough that you make the step. If you already live in the wilderness and kinda despise its vulgarity, while secretly wishing to get more civilized, this may be your path.

While Zen is jumping into the water and swimming blindly, desperately, having no idea what you're doing, with Master metaphorically beating you on the head with the stick of prajnaparamita repeatedly as you try to turn back to this shore, until your pain instinct pushes you in the only direction away from pain - and you find your Master waiting for you smiling and hugging on the other shore. Then in post-realization you catch up on the theory as a communication tool. If your world is in its decadence phase and getting hit on the head is your idea of fun then Zen may be it for you.

So the two traditions are pretty much at odds with each other. The better you are at Madyamaka, the more your Zen Master will beat you - but your frustration will not mature in gidan as you will seek comfort in the certainty of Madyamaka, while at the same time finding it absolutely useless for your Zen plunges into the uncontrived emptiness. I may be wrong, but this is what my logical mind tells me.

Or it may actually work better this way, provided you have equal faith in both teachings and equal connection with both the live teachers - and never directly cross-question one's teachings with the other but skilfully relate and synthesize them in your head. But in practice most likely you will either spill one's wisdom to the other and draw their anger at you, or end up dropping one and going with the other.

A more traditional way would be to study Madhyamaka for several years and then make a clean cut to Zen. Do as I say, not as I've done. Gosh, am I getting old?