You typically don't have to connect to original carrier to get updates but it is up to the original carrier to setup the restrictions. For example, AT&T version of Akpha's Lollipop update is restricted to those who have AT&T sim in the phone as well as connected to AT&T network in US. I suspect that has a lot to do with the size of the update, about 1GB of download, so that AT&T doesn't have to waste its server resources for non-AT&T customers.