"Pushing" may be used in a physical sense: "The panicked crowd pushed over anything and anyone in its way." It may also be used figuratively: "His father pushed him to study harder." The verb "persuade", however, may only properly be used to mean to change someone's opinion or decision (including a decision to take, or to refuse to take, some physical action), by means of argument, threat, or reward. The key is that persuasion is not used to physically force someone to change location or action, but rather to indirectly cause a relocation or an action by changing the will of the person being persuaded.
Your example, "Pushing them off to the side.." seems to be a non-physical situation where "them" is involuntarily being metaphorically relocated. They ("them") aren't being persuaded to decide to move to the side, but rather forced "to the side" against their firm will to remain.
"Off" in your example is simply an adverb that indicates (barely) where "them" has been pushed. "Off" may be used with "push" in either a tangible or intangible sense: "He was pushed off the cliff", or "He was pushed out of his job". It is not proper to say "He was persuaded off the cliff". Even in the rare instance where someone is talked into committing suicide, one would have to say, "He was persuaded to commit suicide". In English one can not be
Using "persuade" only to mean changing someone's mind is probably the safest course.