Also, constructs need a source of power. More sophisticated ones can use a self sustaining mana energy, or something similar, but simpler ones are typically powered by a power source, like a gem, or core designed for the construct. If you know what that form of animating power is, then it would be easy to destroy or nullify, thus making anti-magic a good tool against it. Then again, were it warded, you would have to first breach those, as I have no doubt you can ward against dispel and anti magic in this RP, as both are forms of energy.