The stress of the move has triggered something- and it sounds to me like herpes related conjunctivitis. If a kitten has ever has a herpesvirus upper respiratory infection, even when it is "cured" the virus lays dormant in their system. When stress occurs it can pop right back up. Conjunctivitis makes the lining of the eyelids very red, swollen, and weepy.
In the grand scheme of things it is a small problem, but one that definitely requires veterinary attention. Eye problems hurt, and if he rubs his eye at all he can cause an abrasion on his cornea that could permanently affect his vision.
Most problems get more expensive if the treatment is delayed, not less, and with eye problems you do not want to fool around. Go to the vet and get it under control before it gets even worse.
And there is a bit of a silver lining here- when he gets home from the vet he will be so glad to see his familiar space he will be happy to be there!
