The process is a little obscure. Our source for this is Appian, and he's not always the sharpest. In fact, Mouritsen thinks Appian is basically 2nd century CE historical fiction, with little relationship to actual events in the 1st century BCE. Most historians aren't that distrusting of him though.

For those not familiar, Rome had several assemblies all male citizens participated in, which were organized in different ways. The Centuriate Assembly was primarily responsible for electing magistrates. This was organized into wealth classes called "centuries" that each got one vote. The centuries were distributed such that the two wealthiest classes controlled the majority of centuries. The Tribal Assembly, on the other hand, was responsible largely for making law was organized into 35 tribes which derived their members geographically. There were four tribes for people who derived their origin from the city itself (Urban tribes) and 31 "Rural" tribes for Romans living elsewhere in Italy. Each tribe got one vote. While wealth apparently isn't a structural issue in this assembly as it is in the Centuriate Assembly, since Romans had to vote in person, the Rural citizens who could afford to travel or live in Rome were the ones that could take the most advantage of the vote. Romans were very aware of the structure of this system and did things to make sure the vote was controlled by the right people. For instance, a freed slave necessarily became a citizen, with a voting century and tribe, but to limit their influence on the state they could only be enrolled in one of the four Urban tribes, making their individual votes almost worthless.

Ok, on to the Social War. The reasons for the Social War are disputed: Rome's allies revolted for Rome either to win the Roman citizenship, to found federal Italian state, or to destroy Rome and the Empire. The traditional narrative derived largely from Appian and basically unquestioned until Mouritsen (1997?) is that Rome made increasing demands on the Allies, who had little input in policy and saw little rewards from that policy. In 91 a number of these Allies revolted, some remained neutral, and some (the Latin allies) fought actively for Rome.

In 90 the Etruscan and Umbrian communities had not yet taken arms against Rome, but apparently both the Romans and Italians thought they might go over to the Rebels. One of the things Rome did to forstall this was to pass a law enrolling allies who had not rebelled in the Roman citizenship. This is what Appian says:

They sent this decree around among the Etruscans, who gladly accepted the citizenship. By this favour the Senate made the faithful more faithful, confirmed the wavering, and mollified their enemies by the hope of similar treatment. The Romans did not enroll the new citizens in the thirty-five existing tribes, lest they should outvote the old ones in the elections, but incorporated them in ten new tribes, which voted last.

As part of the settlement of the War in 89, Rome extended citizenship to belligerent Allies if they surrendered, which most accepted:

Each body of allies was enrolled in tribes of its own, like those who had been admitted to citizenship before, so that they might not, by being mingled with the old citizens, vote them down in the elections by force of numbers.

This is confusing. The previous citizens must mean the Etruscans and Umbrians, but they weren't each enrolled in a tribe of their own, but rather distributed across ten tribes. There were also many more Italian communities than 10; each one getting a tribe would multiply the Italian influence instead of, as Appian says the purpose is, restricting that influence. I think it's fair to say the majority of historians think these Italians were enrolled in the same 10 tribes as the Etruscans and Umbrians.

This plan immediately became an issue. Appian says the tribune Sulpicius wanted to distribute the Allies across the 35 tribes in order to use their votes to control the assembly, and some Allies bought into this. Other Romans still opposed enrolling the Italians across the tribes and there were riots in the forum over this issue:

The old citizens saw this and opposed the new ones with all their might. They fought each other with sticks and stones, and the evil increased continually, till the consuls, becoming apprehensive, as the day for voting on the law drew near, proclaimed a vacation of several days, such as was customary on festal occasions, in order to postpone the voting and the danger.

Sulpicius ended up being executed by Sulla for his support of Marius and his legislative work was undone. Getting the citizens enrolled across the tribes remained a policy the Marian party under Cinna pushed for, and there was more violence, but the government was not stable enough in this period to carry out a census. An attempt was made in 86, but that was never completed, and it was only in 70 Gnaeus Lentulus and Lucius Gellius completed a census, and the new citizens were enrolled in the existing 35 tribes, though they were not evenly distributed.

Lily Ross Taylor's Roman Voting Assemblies is the classic study and I don't think has been surpassed, even though it was published in what... 1960? She has a community-by-community breakdown of who ended up in which tribe, if you want that level of detail.