Preserve order of inputs to outputs, while each output comes from a file that should get read once

I'm struggling to come up with the best way to structure my program, regarding a specific function.

Here are some details:

• The input to the function is a vector of (latitude, longitude) coordinates
  • These can be in whichever order the user supplies
• The output of the function should be the elevations at the input coordinates, in the same order as the inputs
• The elevations are retrieved by accessing pixel values in TIFF files that are named by whole number latitude, longitude coordinates, e.g. n40w106, n40w107, ...
  • This number represents the top left corner, for example:
    • (39.5, -105.5), (39.8, -105.2) would both be in n40w106.tif
    • (40.3, -106.1), (40.5, -106.9) would both be in n41w107.tif
  • This means that multiple points in different spots in the input vector could need to be retrieved from the same file
• I already have the all of the functionality to take input points and retrieve the elevation from the corresponding files

What I am struggling with is these extra requirements that I want to ensure:

• Each file should only get loaded once
• Only store the file's data in memory if it is being used currently (so I don't think caching will work)
• The order of the outputs should correspond to the order of the supplied inputs

Is there some existing paradigm or concept in CS that achieves what I want?

I'm also interested in if there is a model that works well with concurrency. For reference, I am using the Rust programming language.