Falcon Knight

Regardless of the name, the Falcon 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 to other character classes. Their ability to use swords boosts combat efficiency in the sense that (in weapon triangle Fire Emblem titles) they have no real weapon weakness.

Combat
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.

Promotions
Falcon Knights, being the promoted versions of Pegasus Knights, can not be upgraded any further than they already are, except in Fire Emblem: Radiant Dawn where they can be promoted into Seraph Knights.

Fūin no Tsurugi

 * Yuno - the eldest sister of Thany and Tate as well as wife to Zealot

Sacred Stones

 * Syrene - commander of the Frelian army's 3rd battalion and older sister of Vanessa

Path of Radiance/Radiant Dawn

 * Tanith
 * Sigrun
 * Marcia(Radiant Dawn only)

Trivia

 * In Fire Emblem: Shadow Dragon, it is possible for a Pegasus Knight to promote into a Falcon Knight with an Elysian Whip. However, the Elysian Whip is sold only in the Armory accessible via Wi-Fi Connection. Otherwise, the Pegasus Knights will promote to Wyvern Rider.
 * In Fire Emblem Gaiden Falcon Knights seem to possess a predecessor to the skill Slayer as they do triple damage to monster type enemies.

Also See

 * Pegasus Knight
 * Seraph Knight