I am having the same problem with the same setup: a 2011 Time Capsule and an Airport Express. I've had my Aiport Express hooked in for about 2 years, so I know it is not the problem. The only difference I have is the DHCP will stop working on the ethernet, but still work over WiFi from the TC. Rebooting the Time Capsule fixes the DHCP issue.
I am running version 7.6.4 in the Time Capsule.