Population estimates prior to accurate census are always problematic, but as /u/kumbhira touched upon, most of scholars would agree that Kyōto ended the Sengoku (1600) with a larger population than it started with. It's not clear when exactly the city returned to its pre-Ōnin level, but thanks in a large part to the patronage of the unifiers, it definitely ended the Sengoku as one of the leading (if not the leading) population centers of Japan. In addition, it's been demonstrated that the destruction is limited to the temples and palaces area of northern Kyōto. Southern Kyōto, the districts for the merchants and artisans, escaped the conflict relatively unscathed. Southern Kyōto was both not the focal point of the conflict, and provided essential logistics support for both sides, and so the armies spared the area. So if you were a commoner you might not have lost your home and livelihood at all.

Assuming then, you lived in Northern Kyōto and survived the fighting, the burning, the famines, and the epidemics of the Ōnin and early Sengoku Kyōto. What then? Yes, then a lot of people decided to leave Kyōto. Ironically, one such group of people were the daimyōs themselves. The Bakufu originally had mandated all daimyōs not in the Kantō, Kyūshū, and the North East to live in Kyōto. Some of these left Kyōto as part of the negotiated peace settlement. Others left to expand the war to the countryside against members of the opposite faction. Many more left to take charge in their provinces. Whether it was because it was turmoil in their provinces that originally contributed to the outbreak of the Ōnin War, or the prolonged war began to cause unrest in the provinces that were mobilized for too long with nothing to show for it, the war was causing many of the shugo to lose control over their own provinces, and some of their deputies were turning on their absentee lords, and now these lords had to run back to the countryside in order to try to salvage their control over their own domains. Now not all the daimyōs left. Certainly many, especially the "winners" stayed. But without a doubt a large number of daimyōs left Kyōto never to return. Likewise, many lower warriors, like Ise Shinkurō (Hōjō Sōun) and Nagai Norihide (Saitō Dōsan, though according to the latest research more likely Dōsan's father) also left Kyōto for the countryside, on official orders to help the war effort and/or to try their own fortunes.

Large numbers of the aristocracy also left Kyōto. To be sure, most probably stayed. But the problem for the aristocratic families was that tribute and income into the city had decreased dramatically. The fighting greatly cut into the abilities of daimyōs to send tribute to the center. Similarly, income from estates in the countryside were being gobbled up by locals for the wars and the amount that reached the capital plummeted. This applied to the imperial ones too. The imperial household became so poor that all court functions were kept to the bare minimal required, to the point that no official empress was named for most of the Sengoku. Less court function means less work for the aristocrats to make a living from. So many aristocrats also went to the countryside. Some went to run their manors. Some went on behalf of the court or their clan to try to get tribute from the daimyōs. Some sought the patronages of the daimyōs, function as teachers of art, culture, etiquette, literature, or diplomats. A few, like Konoe Sakihisa, even went looking for a lord who could restore order to Kyōto. Similarly, artists, poets, and courtesans also followed the trend looking for patronage among the major local daimyōs rather than the impoverished aristocrats in Kyōto. Aristocratic families also began to marry their daughters to major warrior clans. The marriages allowed the daughters to live and secured the families valuable resources to continue their survival in exchange for prestige and a tie to the capital and court. Takeda Shingen's wife was of aristocratic birth. Jukeini, who wielded considerable political power for his husband Imagawa Ujichika, then son Yoshimoto and grandson Ujizane, also began life as an aristocrat.

As previously touched upon, by the time of the unifiers at the latest, the trend reversed, and people from all of the above groups flooded back into the capital.

So to sum up, after the dust settled, most people still decided to remain. But a large number of aristocrats and warriors, and people with ties to them and relied on them for livelihood, left Kyōto for their estates/domains in the countrysides, or to political centers in the countryside to try to seek out a living with the local daimyōs.

2.000 sounds too little, since it wouldn't able to maintain city functions like constructions, distribution and temple managements. According to "Four Thousand Years of Urban Growth: An Historical Census" by Tertius Chandler, he estimates the population of Kyoto in 1467 before the Onin War, as 150,000, from the writings of "Chronicle of Onin" which describes the number of homes being 30,000 (so I guess 5 residents per home). He also estimates the population had reduced to 40,000 in 1477, but I cannot find how he came up with this number. Meanwhile, Tomohiko Harada estimates the population of Kyoto would have been recovered to around 100,000 by1500, from the writings of "Gojiganin Denki" written at the time.

​

But anyway, "where would you go" is just a guess. You could go outside Kyoto where your relative might live, or be a refugee in the safety of temples and camp out, or stay in the surroundings and camp out in wild, just as we will do if we are in the same situation.