Neither function governs empathy, and nor do any of the other functions. The functions are more about how a person reacts to what their empathy is telling them, but the theory does not dictate a person ability to feel for others, or the abscence of that capacity.
Fe is to the urge to respond to social practicalities and opportunities. It is proactive rather than contemplative.
Fi is the urge to reach opinions based upon ideas of social, moral or emotional worth rather than just practicality. It is contemplative rather than active.
This will effect how people respond to what is happening around them, but is function theory does not and cannot predict an exact response on it's own. For example, an Fe user who see's someone in depression may well decide to try and help them... but they could just as well decide the person is weak and vulnerable - the perfect chum to take advantage of. Both fall under the broad Fe spectrum of social necessity/opportunity and both take a proactive approach to the situation.
An Fi user who see's the same problem might decide that it is the right thing to help that person, even if they don't know them. They may do this simply by sending a little time with them, acting natural to create a friendly, stable environment rather than directly addressing the issue. This gives the person somewhere safe to sort out their own issues. They might also decide that the person is going to bring pain and misery to all around them, and is thus best avoided all together.