If there is a folder called "Document Fonts" in the folder where you place the InDesign file, it uses the fonts in that folder. If there are fonts in that folder that have the same name as fonts installed in your OS, this can cause font conflict problems.
If you go to View -> Display Mode -> Normal, do you see pink highlighting? That means that the font you were using is not present, and InDesign is using a fault installed by default with InDesign, like Minion Pro or Myriad Pro, instead of the missing font that you intended.
In other words, it's not changing its relationship to the baseline - you are most likely accidentally rendering it in a different font when you move the file from one place to another. I'd guess that it's using a font in the Document Fonts folder before you move it, and dropping to Minion after you move it to a place with no Document Fonts folder.
Thanks Joel!!! That was it.