I am DMing a small hunger-games-like adventure where there are multiple parties of NPC who also are participating. It is pretty likely that the PCs will happen upon some of the NPCs fighting some monsters.
How do I handle the PCs joining the fight, do I roll initiative all over again? Do I just roll for the new combatants and add them to the round?
Also see this similar question for Pathfinder:
How to handle some new NPCs who enter a fight in progress?
In general it's reasonable to have the new combatants only roll initiative and just add them into the round.
One thing that MIGHT change how to handle the initiative is if all or several of the PCs decide to act together at the same time. For instance, two archers could decide to both shoot at the same time. They could use the same initiative then, representing the faster one waiting for the slower one. "You ready? Shoot on 3 . . 1 . . 2 . . 3!"
To answer this question, Let's look at the rules, PHB p.189:
Combat Step by Step
1. Determine surprise. The DM determines whether anyone involved in the combat encounter is surprised.
2. Establish positions. The DM decides where all the characters and monsters are located. Given the adventurers’ marching order or their stated positions in the room or other location, the DM figures out where the adversaries are—how far away and in what direction.
3. Roll initiative. Everyone involved in the combat encounter rolls initiative, determining the order of combatants’ turns.
4. Take turns. Each participant in the battle takes a turn in initiative order.
5. Begin the next round. When everyone involved in the combat has had a turn, the round ends. Repeat step 4 until the fighting stops.
So, you've done this and fought a few rounds between group A and group B when group C comes along.
1. Determine surprise.
No one in either A or B can be surprised because they are aware of a threat (A is aware of B and vice-versa). They may or may not be aware of C but that doesn't matter. On the other hand, members of C can be surprised depending on the circumstances, however, it is unlikely that all members of A and all members of B are "trying to be stealthy" so opportunities for C to be surprised would be rare.
Notwithstanding, it may be reasonable in some circumstances to allow the members of C to begin the encounter hidden; in which case they make a Dexterity (Stealth) roll and follow the rules for the Hide action. For example, if C is stealthily advancing towards the sounds of battle or C takes the opportunity to hide and wait for the battle to come to them.
2. Establish positions.
The positions of A & B are already known so you just need to work out where C is. Essentially this is no different from what you did with A & B in the first place. The location of C will depend on local geography and if C went to the battle or the battle went to C.
3. Roll initiative.
The rule here is really simple:
3. Roll initiative. Everyone involved in the combat encounter rolls initiative, determining the order of combatants’ turns.
Well A & B have already done this so C just does this and slots in wherever they fit.
You do this when C becomes (or can choose to become) involved in the combat. As a rule, do this when C is close enough to the battle that they could intervene (or be drawn in) on any creature's turn (in group A, B or C). On a big flat plain this would be when a single turn could put a member of C within range of A or B's ranged attacks or vice-versa. Ina dungeon, this would be when a single turn could put C in the line of sight of A or B (e.g. by opening a door, moving down a corridor, entering an illuminated area etc.).
4. Take turns.
C is now slotted in and they each act when their turn rolls around.
If the members of C wish to intervene simultaneously then they can use the Ready action to say, for example, "I cast my spell/use my bow etc. when the last member of team C takes their action."
If C is hidden, then only members of A & B who perceive them can act against them.
5. Begin the next round.
And we're done. You now have the big battle your heart desired.
I think this depends on how you are handling the combat that they came across. I think there is three main situations you might come across.
The party opts to use something as a trigger to act (e.g. one group wins the combat, then they attack). In a situation like this I would re-roll initiative (since its effectively a new combat).
The party immediately attacks, I would just roll them into the current initiative next round.
In a situation where they enter a battle unaware (see this for more ideas on this: How does surprise work in D&D 5e?).
Also you should be aware of this - Can I delay my turn in D&D 5e?. I would keep this in mind when entering them into combat - since I think effectively this means that when they enter they all have to roll initiative and keep with what they roll in terms of the order.
And as a side note, always remember that you can use your discretion as a DM to bend the rules to better suit yourself or your party. I sometimes like to give a surprise round of combat where everyone gets a single action, or give negatives to initiative if a monster was busy/not paying attention.
When I run an encounter I have all my NPCs - even those not in the initial combat scene but close enough or hidden to come in soon - rolled into my initiative sequence so that when ever I need them to come in they can fit right into the turn order.
I also have initiative rerolled after massive events that affect every one, like walls falling and making a large rumble, or the whole area catching fire, etc.
