How to approach Professors from other institutes (without potentially offending the ones at mine) tl;dr - I want to rely on Tutors from other institutes without offending the ones at mine (for not turning to them instead)



I'm studying Undergrad Medicine in South Asia. 

The curriculum is pretty interesting, however, there are a few problems I face.

Most of my Tutors/Professors,


Aren't exactly keen on discussing topics/information that isn't explicitly mentioned in our course textbooks (which in turn, is the work of local publishers and is regrettably riddled with poor wording, shoddy formatting and incorrect or outdated info)
Are usually reluctant to provide us with the rationale/intuition, even for stuff that is in our textbooks.
Are usually very busy. So it gives me next to no time to sit down and clear questions with them. (Texting or mailing them is just as difficult)


They're all really nice people and have been very encouraging so far, but for the reasons listed above (exacerbated by a ridiculously rigid education-system), I've found it difficult to grow as a student under them. Changing courses isn't an option I have.

Therefore, I hope to approach (E-mail?) Professors from other institutes with my questions. Assuming they're able to reply reasonably quickly (and satisfactorily), I'll seek to establish regular interactions with them.

However, there's a very real chance that they are in contact with the faculty at my institute (among others); my institute regularly holds Conventions to which faculty-member from other institutes are invited. If my Tutors were to find out that I'm actively turning to people other than themselves for my questions, they're likely to take it as a personal insult.

So on one hand, when I go Tutor-hunting, I need to make it clear in my E-mails that the resources and Tutors at my home-institute are inadequate (regrettably) for my growth as a student, lest they misjudge the gravity of situation.
On the other hand, I can't just bluntly tell them "My Tutors and I aren't exactly on the same wavelength" (this is putting it mildly), because should they end up meeting my Tutors and mentioning this, it'll cause irreparable damage to my relationship with them.

I suppose my question ultimately boils down to this:

So when I first approach a Professor from another institute (this is likely to be by E-mail), how exactly do I word my introduction? 
In a way that won't potentially incriminate me down the road, that is.
Most important, I question your statement that you need to "make it clear" that you are dissatisfied with your current faculty. That seems like a terrible idea unless you wish to break ties with your current institution. 

You can, of course, reach out to people about topics of interest without giving motives beyond your interest in their work and the answers they might give you.

But there may be little incentive for people to give you very much support, as they are busy and have their own responsibilities. But I don't think you improve that incentive, at all, by complaining about your current situation. 

If your emails are subject/topic based then word getting back to your current professors should have no consequences. But if you say offensive things, you can expect that people will take offense. 

But you might also ask yourself whether you are at the right institution or whether you should be looking for another.