Hi there...
No this is not normal. It does sound like cat flu which he may have been carrying before you got them vacinated (vacs are ineffective in a kitten already carrying cat flu). Kittens can carry cat flu but not succumb to it, but when faced with a stress factor ie, new home the stress can bring out the cat flu
I would highly recommend a vets visit, just in case it is cat flu. Cos he will need antibiotics (to make sure it doesn't go to his chest/lungs) and some antibiotic eyedrops to stop ulceration occurring.
On the other hand it could be an allergic reaction to something in kitties environment, though this is rare... Chlamydia is something else to consider as well, it causes heavy discharges from eyes,nose etc.
Wishing you and your new kitties a loving and long life together