Use a PvP conflict as a cliffhanger for next session, make the minority group NPC enemies and allow their old PCs to make new characters to be immediately included.
While my previous answer stated that it's not possible, this answer uses "4e" and "intraparty conflict" as a given, and then posits the campaign that would result. While this is from my personal experience, it'll require very good players to pull off well.
The core idea (taken from when one of my characters became a solo for the rest of the party to fight) is that the players are very separate from their characters. This only works if high party turnover is OK.
Assuming that you have a group that is comfortable with frequent rerolling of characters, you can absolutely enable intra-party conflict with some house meta-rules. Set out at the beginning of the game that: whenever any of the party draws weapons against the others, the minority in the conflict will be turned into NPCs and fought as elites or solos next game.
In a sense, the various feuding powers of the world can for whatever plot-related reasons, vastly empower "treasonous" party members. (Make sure you have a good reason for this, as this empowerment must not violate suspension of disbelief). So long as the party has an odd number of people, whenever there is a "drawn weapon" a few things should occur.
- A ballot must be cast asking not for "who agrees with the majority?" but "which of these plot lines is more interesting?" By having the conflict be interesting-seeky, rather than consensus-seeky, many future conflicts will be spawned.
- Spend whatever remains of the evening fleshing out the faction of NPCs that "just spawned," their narrative resources, and the new characters.
- Write down the conflict and the stakes for the start of the next game. By opening the game with a description of the major battle that's about to occur (and whatever narrative manoeuvres were necessary for it to happen) and the player-agreed stakes of the fight, these sessions will open with a big bang.
- For the most part, don't have battles be to the death. Almost all conflicts should allow for different tactical end states such that factions are jockeying for power with "total war" being a very rare thing indeed.
This will produce a very odd and challenging, but potentially very rewarding 4e game. You may want to tack on REIGN for faction conflict, and have each player also run one of the major factions of the world, to keep the conflicts fresh. Make very stark ideological conflicts in a "dark n' grim" world to keep the moral decisions flowing. Make sure that there is no such thing as a "free" good decision, and that every narrative decision the party makes to help others costs them something.
Let us consider a worked up example of the above:
First, the world needs to be more outré than normal. Let us draw inspiration from the Ultima series as well as some of the very early computer strategy games that had ideals fighting each other. Furthermore, land in freehold exposes an empathic environment and a clustering tendency, to be near other land that shares similar opinions. When opinion changes, the land changes.
Therefore, we have a world of floating mobile chunks of land that correspond in size to their ideological population, an excellent basis for a series of short adventures with a moral basis. Each of the major "empires" should exemplify a given vice (however you want to operationalise them) and its virtues (such that you can actually have a social order).
Let us posit a small kingdom of sloth, modelled after Omelas. We have a cleric and a fighter from there, a wizard from a nearby sloth-anger (mmm, merging) police state, and a rogue and bard from sloth-greed corrupt oligarchy. They're on a mission against the wrath-pride "empire of awesome" when they come upon a moral dilemma (engineered into the adventure, as it's the whole point of the game.)
The slothy lands are lush and fertile, with lots of fruits, berries, and rolling hills. (Why farm when you can just eat grapes?) The wrathy lands are semi-industralized (to whatever extent fits the game) with granite tors and other impressive geography.
As the party journeys and encounters their moral dilemma, the cleric and fighter are obviously willing to sacrifice an innocent on their way to the objective. The rest of the party disagrees. An argument starts, and weapons are drawn. The game halts while people work up the consequences.
The party decides that the minority story is more interesting and because they want to explore questions of "the greater good." They then see what the consequences of the defection are (the cleric and fighter leave rather than coming to immediate blows) and the mission is a failure due to their defection. They decide that the new PCs come from some other land who also buys into the tyrrany of the majority, and the decide the cliffhanger conflict will be with the old characters who have come seeking the defectors with an "enforcement squad".
Here, the group is very heavily involved in building the narrative, which is unusual for a 4e game, but quite possible. The party conflict stops play and aborts the mission, which is fine as few missions are for extremely high "save the world" stakes or have represented a very long commitment. New characters are built to support the new party (and given how much character mobility exists, I'd recommend not worrying about anything but interesting plot-gear and just saying all characters always have level appropriate gear (excepting plot) on them.)
While you lose the experience of playing the same characters from 1-30, this sort of game does allow party conflict and "PvP" by turning it into PC v. NPC when weapons are drawn. The world is rather more grim than the typical adventuring world, but it fits the "moral dilemma" style of play.
It's possible, but it sure will be challenging to engage in.