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Units that have Swiftness (X) have X% more Combat Speed than usual, causing them to move faster on the battlefield. Your other movements aren't boosted similarly, or at least aren't controllable enough to be useful in combat or anything else.
This is not to be confused with Quickness, which provides more useful buffs but generates increased fatigue.
Swiftness is a buff shared between Nature and Air:
For stacking purposes, you should know that the three spells do NOT stack with themselves or each other.