Unfortunately, it did not work: After downloading the update, the phone went into a boot loop, because CWM 184.108.40.206 (which I had installed in order to root the device) cannot flash the update. The phone then boots up, just to shut reboot almost immediately into recovery to try again.
To get out of this boot loop, I manually entered recovery and wiped the "cache" partition.
I retried with the latest CWM 220.127.116.11, this also did not work.
So I had to get the original stock recovery image for the Moto G and flash that. I did not easily find it with a web search, so in the end I downloaded the matching stock SBF image for my installed firmware (in my case "Blur_Version.176.44.1.falcon_umts.Retail.en.DE") from the Moto G firmware page on droid-developers.org, looked into the zip file and found that there is a "recovery.img" in the archive.
Now everything was easy: boot the phone into fastboot mode (power on + volume down), then
fastboot erase recovery
fastboot flash recovery recovery.imgreboot, once the phone is booted select "install system update" from the notification, stock recovery boots and installs the update, done.
I did not even lose root access, so I kept the stock recovery for now.
Later on I checked the md5sum of the recovery.img I flashed and of the recovery partition on the phone and they are identical, so the 4.4.4 update did not flash a new recovery for me. I'll keep the old one around in case I need it again.