Deal with the meltdown before you address bad behavior. This one is critical. My son and his sister fight just like any other siblings, and just like any other siblings; they both know exactly how to push each other’s buttons. Once he starts melting down, he can’t focus on anything else but how angry and upset he is. He has to calm before he can take correction of any kind. I put my arms around him and hold him as I rub his back. I get him a drink of water. I’ve even been known to sing to him, if that’s what it takes. Once he’s calmer, I sit him down, and then we talk about how he shouldn’t have thrown that toy, pushed his sister, or screamed in her face. We also deal with consequences (apologies, loss of privileges, etc.) Trying to have any kind of a rational conversation with an irrational child is an exercise in futility. Treat the meltdown first.
If you’re out in public, remove the child from the venue, if possible. I’ve carried, pulled and yes, even sort-of dragged my children (both of them!) out to the car, leaving a full cart of groceries in the grocery store. I’ve asked a waitress to box my food and told her I’d be back in ten minutes to get it. I’ve apologized to front desk receptionists and asked them to reschedule our appointment. Luckily, I haven’t had to do it too often, once they got used to the idea that Mom meant it when she said we were leaving. Once we’re in the car, they can scream and yell their heads off. Eventually, they get tired of it. Being on display while your kid is carrying on is brutal, and the extra censure you’re feeling (or think you’re feeling) from everyone around you is only going to make you more desperate and not as in control.