How can a PhD student seeking to transfer to another PhD program prove his/her research skills when his job is mostly teaching? Suppose that our friends Alice and Bob are both academicians (academician: a member of the faculty of a college or university) in universities A and B respectively.

University A requires a lot of teaching and administrative work, whereas university B requires a certain amount of research and only a small amount of administrative stuff.

Alice enjoys research very much but she cannot spend enough time to conduct research and be productive in terms of publications.
Bob, on the other hand conducts research since it is part of his job and does not really spend effort to improve his projects.

Let Charles be a member of the admission committee in university C that has an open PhD position. 
If both Alice and Bob applies to the position, he will see that Bob has been involved to more research projects than Alice, moreover, Alice has no publications and has spent her time on teaching.

The thing is, Alice does teaching really nice and gets positive feedback from both professors and students. It is clear that Alice can do her job above-the-average. Bob, on the other hand does not even need a feedback. He proved that he can do research by his projects and publications.

Suppose that both Alice and Bob are started working as faculty when they were master's students and now they are PhD students who want to find a position in a different university because of their own reasons.

Considering that both Alice and Bob are currently faculty members, Bob is one step further from Alice in terms of getting the PhD position, if not many steps.

What can Alice do to prove that she can be a way more better researcher given time and opportunity? 
Is there any "Charles" who is ready to give that chance?

If you are "Charles" (a member of an admission committee), would you give that chance to Alice? If you would, under what conditions?
"What can Alice do to prove that she can be a way more better researcher given time and opportunity? "

How do you know? From your description, Alice pines to do research but has absolutely nothing to prove that she is a better researcher. Teaching and research are entirely different skill sets; being good at one does not mean being good at the other! She has allowed herself to be stuck in a mainly teaching role, rather than a research focused role. Universities don't just want researchers, they want sharks; people that know exactly what they want and go after it. Alice making excuses will not make her appear to be a shark.

If Charles is interested in hiring a researcher, why would he ever choose someone who has, from your description, absolutely no experience in research versus Bob who presumably has a track record? That said, if Bob isn't a shark either but just kind of drifted from lab to lab and acquired the 15th name on a couple of papers along the way, I can guarantee that they aren't going to be too interested in Bob either.

If Alice wants to do research, then she needs to make some fundamental changes in her life. Her best bet is probably to carefully read her contract with regard to responsibility hours then go to her department and demand a change in her schedule to allow her greater time for research. Then she needs to buckle down and produce and publish. 

Alternatively, she can try applying for places and say that she is looking to break away in order to do research, but again research is expensive and she will really need to sell whatever experience she has in order to convince them that they should take the risk on her.

I'm genuinely sorry to have to be blunt like this, but I see no other way. I think there should be far more appreciation for teaching oriented professors in a learning institution. Instead they get some lip service and little else. But you came here for truth, and this is the truth as I know it. Nobody will care about what Alice might do if they see absolutely no background that supports such assertions.

On a personal level as an example: when going for the NSF graduate fellowship, I didn't receive one even though the reviewers stated that the idea was great, the proposal well written, and the research seemed reasonable and doable, because I hadn't published enough papers. As an undergrad, I had published three. That wasn't enough though, apparently. Nonetheless, it hammers home the point that it didn't matter how bright I made the future sound given that my past wasn't up to snuff.