I think it's all about attitudes, here. Maybe the less experienced rider doesn't realize how their actions are affecting you, and the fact they are whining about not being first is ridiculous... I wouldn't want to ride with someone who whines.
Also if something is bothering you, speak up. Say "Hey I would appreciate it if you would wait for me to mount up before taking off." or something on those lines.
Yes you feel frustrated but again speaking up will help. And not in a harsh way as to put them off, but a way that shows them that you are serious.
It's a communication issue to me.
Now if you have spoken to your riding partner and they are still whining and being a pain, then don't ride with them. Simple as that. Somethings have to be learned the hard way.
Why risk your safety if they aren't on board?
"Strength is the ability to use a muscle without tension"