Yes, it resolves all the combat steps indicated in the rules.
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If I eliminate a unit with the switches, do we continue the combat?
Yes, it resolves all the combat steps indicated in the rules.
How does Displace work as a defensive switch? Do you get to move away before the damage is allocated?
You move before the damage is distributed, but despite you displacing or placing yourself and, thus, disengaging, the combat continues and resolves until the end. Displacement and placement switches do not prevent you from receiving damage.