The civilization isn't the people
Yes, the major civilizations can contain the minor ones. They can deny usage of the Jump-Gates, and they can blockade planets from orbit.
But then they are abandoning the people in those civilizations. Billions of souls, and all of their descendants, oppressed by their own governments, unable to live lives to their full potential.
Part of the goal of the major civilization is to give every sapient the right to travel, explore and reach their full potential. Those backward minor civilizations that use this freedom to invade and oppress other planets and even their own people are not acceptable.
K1/K0 wars are not a real war as far as K2 is concerned
Now, small scale wars and disputes are one thing. You disagree about the treatment of some people on another planet? You send soldiers over there to protect them.
Start engaging in weapons of mass destruction? Mass conquest or xenocide? That isn't allowed! Your civilization is sanctioned; reduced gate privileges, complete gate blockade, orbital blockade, and finally other civilizations (and volunteers from the major civilizations) are allowed to invade your planet so your local government can be replaced with one less hostile to the galactic community (and their own people).
The wars of the K0 to K1 civilizations is a kind of "game" to the K2 civilizations. They have rules about how you are allowed to fight. These rules permit massive battlefields and parts of planets turned to rubble, but don't allow slagging of planets or similar.
Even civilizations that disobey the rules are just constrained, and their planets get to be invaded by the rules. Aggressive warlike K0/K1 civilizations use these punishment wars as ways of expanding their own influence - they are permitted to land on the enemy planet and engage in brutal warfare, and the territory and resources they claim from these wars of punishment are partially kept as payment when the war completes.
Interplanetary transportation is limited
A well behaved K0/K1 civilization is permitted control over their own interplanetary craft and allowed to use the jump gates, possibly in a limited area.
If they are less well behaved, first they aren't permitted to use interplanetary craft outside of their internal borders; they have to be "carried" by better behaved civilization's craft. Eventually even internal interplanetary craft use is restricted, and then banned, as their violations grow.
A K0 civilization that refuses to follow the rules might have K1/K2 defended beanstalk on their planet, and they aren't allowed to fly ships into space; ships are just shot down. The area around the beanstalk is defended by advanced technology soldiers. Individuals are allowed to ride the beanstalk (after being searched), as are trade goods - and all interplanetary travel is riding on alien ships.
A K1 civilization that refuses to follow the rules gets bottled up in their system. A jump gate might be deployed and defended in the system; a transfer station where local "well behaved" ships are allowed to dock, and alien ships carry local goods and individuals through the jump gate, might exist if they aren't actively hostile. If the locals engage in war with the K2 jump gate operators, the absolute worst they can do is blow up the transfer station and try to take out the jump gate (isolating them), and even that is unlikely.
Better behaved K0/K1 civilizations are given the freedom of their system, and are allowed to transfer jump gates themselves. The cheap beanstalks and transfer stations still exist, but are less control points as economic resources.
Use of interplanetary craft to engage in warfare gets your entire civilization in trouble. Even Piracy gets you disowned by your civilization (but goes on regardless, unofficially; so long as the scale remains small).
More than one kind of war
So you can have two hostile civilizations flying through a system and not shooting at each other, landing troops on a disputed planet. On the planet they are engaged in a massive war; but orbital to ground weapons aren't used. Both are "well behaved" civilizations, so have their own interplanetary navies. Both engage in piracy against the other using deniable assets.
Observers and mercenaries also contribute to the war - observers are there to make sure the limits of the war are kept up, and mercenaries are there to fight for profit.
Another kind of war would be one where one of the parties is ill-behaved K0 civilization, and the other is well-behaved. The well-behaved party is landing troops in its permitted region and either defending a region, or attacking the other side. The ill-behaved side is denied access to space. Its (legal) imports are also limited, with naturally a thriving black and grey market of weapons technology being smuggled in.
A final, more violent case is a K1 ill-behaved civilization being punished. Here, the K2 and well-behaved K1 civilizations are attempting to punish it. One or more systems of the K1 civilization has a no-fly zone imposed on it, and the well-behaved K1 "helpers" and K2 enforcers are landing on planets and engaging in a war of correction.
Other parts of the K1 civilization may be permitted to continue to operate if they aren't actively attacking the jump gate and transfer stations.
In a more extreme case, the K1 civilization goes completely rogue. It manages to destroy multiple transfer stations and even occupy some jump gates. On the K2 side, the jump gates are isolated from the network, with stuff sent from them destroyed. K2 forces take system after system, including conquering the planets and "correcting" their people, with the goal of completely dismantling the K1 civilization and making it well behaved.
K1 "well behaved" allies fight along side the K2 civilization both in space and on the ground. The K1 civilizations get to exercise their interplanetary navies and troops, and some of the territory they conquer they get to keep (or get rewards from) later.
Categories used
I am using the Kardashev scale.
K0 is short hand for planet-bound civilization, like ours is. K0 is under K1, has less than a planet's worth of energy budget.
K1 is short hand for a civilization between K1 and K2. They have the energy budget of more than all of the energy that a single planet can capture, but less than all that a single star can produce (a dyson sphere or swarm).
K2 is what I'm labeling the major civilizations. They have multiples of star-output energy capabilities as a civilization, but less than a whole galaxy of captured stars to themselves.
Each of these has a power capability (in Watts) ratio of roughly 10^10 between them - a factor of 10 trillion. Civilizations a significant fraction of a K-scale below you are not a threat, except maybe to unguarded civilian individuals.
My prototypical K0 civilization is us - which is about K0.75, up to K0.9, a civilization bounded by overheating the planet from raw energy use. Pre-interplanetary, or at most minor colonies.
The K1 civilizations I'm imagining are K1.0 to 1.3 or so - that is 1 to 1000x the energy capture of a single planet. Interplanetary civilizations, with the vast majority of the civilization not on a single planet, probably bottled up by their lack of FTL.
K2 civilizations I'm imagining in the 2.0 to 2.5 range - 1 to 100,000 stars of energy fully captured, but still far from K3.