There are various ways of putting additional data on a fixed 2-complex and then defining a notion of "discrete conformal equivalence" between different assignments of data, see e.g. the answers to [this question](https://mathoverflow.net/questions/288909/) or the field of [circle packing](https://en.wikipedia.org/wiki/Circle_packing). 

In some of these settings there do indeed exist local transformations which allow you to change the combinatorics of the 2-complex; e.g. one can subdivide triangles in a circle packing or perform [star-triangle transformations](https://en.wikipedia.org/wiki/Y-%CE%94_transform) on certain types of discrete Riemann surfaces, e.g. [isoradial graphs](https://arxiv.org/abs/1012.2955) or ["surfel" surfaces](https://arxiv.org/abs/0802.1617). 

A bit more classically, tilings of rectilinear polygons by rectangles (à la [Brooks, Smith, Stone, Tutte](https://projecteuclid.org/euclid.dmj/1077492259)) can be viewed in this light as well (see e.g. ["Squaring rectangles" by Cannon, Floyd and Parry](https://www.math.vt.edu/people/floyd/research/papers/mageps.pdf)). The rough idea is that rectangle tilings can be constructed from currents and potential differences in a resistor network; the potentials in a resistor network and the currents form a pair of conjugate harmonic functions. Then the classical electrical equivalence moves (including the "original" star-triangle transformation) lead to local transformations of square tilings which change the combinatorics; here are two figures from Kenyon's ["Tilings and discrete Dirichlet problems"](https://doi.org/10.1007/BF02780322).

First, a depiction of the transformations of the underlying resistor network:

[![electrical circuit transformations][1]][1]

And their realization as transformations of the rectangle tiling:

[![rectangle tiling transformations][2]][2]

There are likely more examples. Unfortunately I don't know of an overarching framework which captures this phenomenon (nor even of an exhaustive survey), as this is a rather broad field with influences from conformal geometry, combinatorics, statistical physics, and computer graphics. The references given above are by no means meant to be complete or even representative.


  [1]: https://i.sstatic.net/yy0Lj.png
  [2]: https://i.sstatic.net/aIzIT.png