Ignite and switching weapons

Say I am switching between two weapons. One has no bonus to faster ignite, burning damage or DoT multiplier, the other has them. I deal igniting hit with the first one and then switch to the other one. Will the bonuses from second weapon apply to the dealt ignite?

There is no "snap shotting" in PoE. This means you cannot stack a bunch of beneficial effects, apply them to a duration component (i.e. ignite) and keep the effects after expiration.

If you ignite with one weapon then switch to a second with ignite modifiers, then you should receive the bonus ignite modifiers as long as your second weapon is equipped. I can think of very few situations where this would be an overall effective gimmick though.

That is also assuming ignite modifiers are global. If they are linked to the weapon then you have to apply it with the weapon in order to receive the effect.