IMO, they should. In the real world, they issue the operating system updates. The security updates are the responsibility of the carrier. I think that's because each carrier has their own version of what they put in top of the OS. The only thing I can think of is to use the Software version to find what carrier that it was meant for. Mine is V405UA20G, where 'V' is Verizon, I think. Try to find their security updates to download and install.(if that is even possible) I don't think installing a security update from a different carrier is a good idea. Maybe the XDA forum has information that would help. They certainly have people much more knowledgeable than me.
Good luck.