Not sure that you understand my question.

Doesn't current flow whenever there's a potential difference and a path?

Before the bird even leaves the ground, it should be at ground voltage right? When it leaves the ground and flies to the power line it presumably doesn't undergo a voltage change so still at ground voltage.

The instance it finally touches the wire (which is at thousands of volts), the voltage difference between the bird's head (which was just at ground voltage) and the line is thousands of volts. Thus there is a potential difference (wire to bird) and a path (wire to bird's head) meaning that current should flow until there is no more potential difference.

I'm wondering how the bird get's charged to thousands of volts without getting fried during the initial change in potential the bird goes through. I understand that if the bird is sitting on the wire for some time it's also charged to thousands of volts so there's no reason for current to flow.