A firing line is a group of Soldiers that have lined up. They all aim at the same target and fire on them. This does a lot of damage to Haytham or Connor.
There are only a few ways to avoid taking damage from a firing line. The first is to grab a nearby foe and use them as a human shield. Second is to quickly duck into cover. The third is very difficult: you must charge the line and quickly attack whoever is closest to you.
