In single player (didn't try in multiplayer), if you are slowly shrinking or expanding the border, it stops when you leave the game.

To reproduce:
"/worldborder center ~ ~"
"/worldborder set 10"
"/worldborder set 500 120"
Wait few seconds... (don't wait too long or the border will reach the new limit)
Leave the world and open it again.
And voilà. The border no longer move.