A DANCE WITH DRAGONS

In A Dance with Dragons, can a unit's movement be reduced to 0?

Yes.

Scenario: During a game of Dance - two units (one from each player) are engaged and are both carrying objectives. One unit is forced to drop their objective token due to a failed panic test/being destroyed OR has it taken by an attacker (the other objective holding unit) due to them having more ranks. Where does the objective go (if anywhere)?

The most recently claimed objective is immediately dropped by the attacker – the opposing player may then place the objective within 2” of the attacker (engaged enemies do not automatically receive the token) as per the mission rules. The attacker keeps hold of the objective they were already carrying.