My first thought that I would like you to rule out is that the birds can simply see the rain and fly towards it.
Migratory birds fly at thousands of feet in the air, which means they see the horizon at one or two hundred kilometers. And thunderstorms can even rise to a height of 10 to 12 miles in the atmosphere. Realistically, the limiting factor is light diffraction. A human can identify a storm cloud at a distance of 100 miles.
What prevents a migratory bird, following the air currents and land masses from Siberia and Alaska down towards Australia from visually identifying and flying towards rain a few hundred kilometers inland? How do you immediately rule that out as an option?