To unbalance something means to make it unsteady and likely to tip over. Her whole body began to buckle, unbalancing the ladder Don't lean in -- you're unbalancing the horse
If something unbalances a relationship, system, or group, it disturbs or upsets it so that it is no longer successful or functioning properly. The interplay between the new politics and the modern media will unbalance the political process and inhibit its workings. = destabilize