There's no DoT weapons save for end of the game singular weapon, desintegrator which eats thro armor.
My idea would be fire and few weapon ideas that could do this kind of damage, mechanics and ways of preventing it. It would chew thro hull, if an incendiary projectile deals hull damage it will cause fire that deals unresisted by armor hull damage till it's put out, the fire would scale with the amount of hull damage was dealt, refreshed with every hit that was dealt by fire causing projectile and increasing the dot if the hull damage was higher than the previous one but it wouldn't stack (makes no sense unless there would be a balancing reason for it). Fire would also stay on a phase ship if it was lit so dumb immortal ships can have a lil counter to them. Fire would be fixed by "repairing" it so more reasons to pick the Automated Repair Unit hull mod and perks related to repair. Fire could incentivize possibly taking the damage control elite skill. Also, flame particles would bypass shields, not whatever is carrying the fuel for the flame, just the particles themselves with thicker shields reducing the chance for them to bypass (so for example 1.0 shield doesn't protect but a 0.6 has 40% to block them and the blocked particles would do very low soft flux damage), at least that's the idea. Now to weapons, I'm horrible with names so just the general idea of what the weapon would be will be described.
Flamethrower: extremely low range ballistic weapon shooting just the flame particles, it would chew thro hull but do nothing to anything else unless it has enough damage to go thro the armor, it would also go thro shields as it's just the flame particles, also the reason why the extremely low range.
Incendiary shells: basically bullets carrying flammable load, the hit on the shields would do low hard flux damage, medium armor damage and normal hull damage. On shield hit they would break and release flame particles in a spray pattern like sabot missiles and those would have a chance to be blocked by sufficiently hardy shields. Perhaps breakable mid-flight like a missile?
Flame lance: a beam that does no shield damage, low armor damage and normal hull damage but have the benefit over flamethrower that it has longer range and perfect accuracy.
Incendiary missile: deals as much shield damage as a dud from other missiles, so very weak. If it reaches the ship, drills into it dealing medium armor damage and explodes applying the fire dot. It would also explode with small aoe if broken mid-air, igniting anything nearby but dealing no armor or shield damage when doing so, only hull.
Incendiary fighters: equipped with flamethrowers and breacher missiles, breachers to deal with armor and flamethrower to exploit the lack of it.
Bonus mechanic, there's icy weapons from doritos, they are already balanced to what they are supposed to do but what if they could be changed so they jam weapons they hit, supposedly "freezing" them or shoving ice?, would be thematic.
EDIT: I also forgot, wanted to propose together with all this fiery stuff a new hull mod, something with extinguishers for fire prevention, a logistic hull mod that increases the speed fires are being "fixed" but it wouldn't help with other repairs like damage or emp. For a secondary effect to not be completely useless like many logistic hull mods have, it could provide like +10% to flux disipation speed with a reason that... don't know, cooling in the ship helps with flux?