So, say I had a simple line chart with dots on one part of the chart, and I had an identical one with the dots in different positions. Is there a function in After Effects (or any other Adobe program) that could automatically animate the dots going from there position in the first image to the next?
Nothing in AE is automatic. Something has to be keyframed. In your case you could use linear() expressions to blend between positions based on a keyframed slider that controls the blend amount.