Do you mean the core or do you mean an insert pulls out of the door that has the core in it?
If your core is falling out then you no longer have a retainer clip on the back of your lock's core. Ask a locksmith for one, just ask for a truarc clip or something similar.
If an insert is pulling out of your door than you have a "530 style" weiser, just your normal keyed lock on one side with a locking button on the other. One possibility is that your backset rod is caught on something or maybe something has jammed ( who knows? I'm not there to look
) it into place. either way, to remove the core for one of these locks you need to pull on the bolt rod, insert your key and turn it to a 9:00 (? maybe something else, I can't recall to be honest) position. your lock insert should come out.
The most likeley thing that has happened is during a rekey or something similar someone did not pay attention while re-installing your lock insert.
You need to pull your backset rod down into the handle again (you will have to remove your lock from the door ) and then take your lock and key, insert them into the handle and rotate the cylinder until it falls into the knob. release the rod,Remove the key, play with the handle a bit , lock it, unlock it etc etc, to make sure it doesn't fall out again.
presto. Hope this helps.
