Why did Lucifer care about Mammon crossing over from Hell to the real world? Before coming up to drag John (literally) to Hell, he makes a detour and sends his son back to presumably prevent Hell on earth. Why would the master of Hell not want to expand his domain?
From the script
Mammon has been trying to climb out of his father's shadow for eons.
And this time he might just make it.
(unconvinced) Because he's got the Spear?
And the bridge.
... so how's your son?
And why would that matter to you?
... he's topside...
He's helping your son create his own Hell on Earth...
Well, boys will be boys...
He has the Spear of Destiny.
Satan didn't know that. Now he's concerned. He studies this dying man.
It's not really clearly spelled out, but it seems as though Satan's pride won't let him have anyone outdoing him or anyone else creating hell on Earth - not even his own son.
When Angela Dodson visits Constantine in his home, then leaves and Constantine follows her (approximately 36 minutes into the movie), he says the following:
Constantine What if I told you that God and the Devil made a wager? A kind of standing bet for the souls of all mankind
Dodson I'd tell you to stay on your meds
Constantine Humor me. No direct contact with humans, that would be the rule. Just influence, see who would win.
Dodson Okay, I'm humoring you. Why?
Constantine Who knows? Maybe just for the fun of it, no telling.
When Lucifer comes along later on and saves Dodson from Gabriel, he says
Lucifer This world is mine, in time.
So two reasons here: He wants to win his bet with God, and he wants to stop Mammon from taking over the world, which he considers to be his.