Regardless of the name, these knights still fight from the backs of Pegasus. After upgrading from the Pegasus Knight, The Falcon Knight receives slight boosts to their defense and constitution, but these stats can still be considered below average in comparison other character classes. Their ability to know use swords boosts combat efficiency in the sense that (in weapon triangle Fire Emblem titles) they have no real weapon weakness.
Overall, these characters will still fight and support their allies in the same way they did in their Pegasus Knight forms. Arrows should be feared, but the ability to close in on archers from a long distance can easily be utilized to deal with ranged units. Falcon Knights are a bit more resistant to magic, but avoiding mages is still recommended. If you can use other characters to remove any archers or mages, it wouldn't be a bad idea to do so before sending in Pegasus Knights.
Falcon Knights, being the promoted versions of Pegasus Knights, can not be upgraded any further than they already are.