I'm happy that you have asked me to assist you with your child and hope that you can benefit from my many years of helping parents like you.
I suspect this is mostly due to his seborrhea which will cause the hair to fall out. Kids with eczema are much more prone to seborrhea (cradle cap) and when the greasy scales come off the hair comes with it.
To stop this I suggest you first try Sebulex shampoo for a couple of weeks and if that doesn't work then the next shampoo to try is Nizoral 1% shampoo, which is available without a prescription. I'm sure you chemist can get it for you.
I see you are still offline so I'll exit chat but still be standing by when you return and if you have questions I'll be notified.