Improving "behavior" (I would say "fit" -- the problem can often be on the company's end as well as the employees, and if you want to take the family analogy even semi-seriously, the employer needs to be able to be introspective enough to recognize this) needs to be taken just as seriously as "fire only as a last resort" in these cases; and if you don't have a credible plan to improve fit, you are at the last resort.
(Lots of places that give lip service to an ideal like this, especially places that are still afflicted by heavy bureaucracy, take it the wrong way, and it amounts to "never impose negative consequence and just try to sweep any performance problems under the rug as long as possible"; that's at least as bad as "fire at the first sign of trouble, and never try to understand what went wrong and how it could be improved".)
It's alright to fire as a "last resort", but make the process to come to that decision a swift and confident one - your other employees are watching.