I'd say that this is in fact most of what mathematics is. Seriously! In Linear Algebra we learn about things like Matrix Decompositions (QR, LU, Cholesky, Rank, etc). In calculus we learn about change of variables and Jacobians that allow us to convert from one space to another. We also use change of variables frequently in statistics and metric theory (in ML we have Normalizing Flows). In proofs we often abstract to the opposite of a problem and prove that the opposite doesn't work. We do this because we can't prove the thing we want directly. In optimization we have dual problems where we convert a problem to an easier one. We do this frequently in algorithms and most motivating examples for dynamic programming are doing exactly this. Topology is the study of looking at a mug and doughnut and calling them the same thing. Similarly in knot theory. I could seriously go on and on.
All over mathematics and problem solving we frequently find many techniques of converting the problem we are trying to solve into something easier. I'd even argue that this is what mathematics and most of science is in general. After all, everything we are doing is an approximation. We can't solve the universe, but we can dictate what we see with a carefully laid out language (physics) and use that to describe what we see and make predictions based on it. Everything is just a model and a model is a map. (I'd go as far as arguing that we do this with language, not just in analogies, but in so far as saying that the existence of language itself is a map from a difficult and intractable problem to an easier one)
The thing though is that not everything can be mapped to anything else. That's the real hard part. For example you would not have a bijective mapping from tic-tac-toe to chess and you can prove this by looking at the number of game states that each contains. TTT to chess would clearly be a non-injective mapping.
So it is hard to give you a tip into specific forms of research without knowing more specificity. I would encourage you to see the world like this. This is why many of these fields will encourage you to look at problems through different lenses. Why you'll often see many of the big breakthroughs in fields are connecting ideas from other fields (Nash and Einstein are notable and relatively recent examples). There was Terrence Tao's post about how to solve problems on HN just the other day (and many references to Polya's book) and I'll say that you will find the same recommendations there.
I don't know if there is any specific field that covers this topic but rather I think every field _is_ this topic.