li_at is the cookie LinkedIn sets in a browser that has signed in. Hand it over and you connect the account without ever typing a password — which also means without meeting any of the checkpoints a fresh sign-in can raise.
That is the reason to use it. If an account keeps asking for a verification you cannot answer, pasting a cookie from a browser that is already past it is the way through.
The user agent is not optional
LinkedIn ties a session to the browser that created it. Send the cookie with a different user agent and the session is often dropped — sometimes immediately, sometimes an hour later, which is worse because it looks like a flaky integration rather than a mismatch. Copy the user agent from the same browser you copied the cookie from.
How long it lasts
A session cookie is not permanent. LinkedIn expires it on its own schedule, and signing out of the browser you copied it from can end it immediately. When it goes, the account stops working and the status reads as needing credentials again — which is a reconnect, not a bug.
A cookie is a live session. Anyone holding it is the account, with no password and no second factor in the way. Treat it exactly as you would a password, and sign that browser out when you are finished with it.