After experimenting with the free multiplatform tool GPSBabel (http://www.gpsbabel.org/) and with the Python module Fitparse (https://github.com/dtcooper/python-fitparse), I can share the following:
If you want latitude and longitude only, GPSBabel is fine. It converts the spatial coordinates in a Fit file into GPX format, which Mathematica can import. Unfortunately, it doesn't bring over anything except time stamp, location and elevation, and Mathematica then tosses the elevation data. This was not adequate for my purposes, but for some people it may suffice.
Fitparse worked much better. It requires Python 2.7. I wrote a python script that uses Fitparse to convert a specified Fit file into a text file containing a nested list of Mathematica
Associations, which can then be imported into Mathematica. Using Pythonica you might be able to do the conversion directly, without saving the converted data to disk and importing it, but the save-to-disk-and-import method is no great hassle.
FIT is a flexible format that will contain different fields depending on what is being measured and by what kind of device. My conversion preserves everything in the original file -- latitude, longitude, elevation, heart rate, cadence, temperature, whatever. My tool stores this in a nested list of Associations, where the outer Association specifies the time stamp and the type of record, while the inner list of associations specifies the data associated with that time stamp and record (heart rate, cadence, etc.), and the unit for each one.
You can see my python code for doing the conversion and sample Mathematica code that uses the output, along with a sample FIT file, at github.com/MichaelSternNYC/fit2Mathematica.