Some local analysts quoted by RFE/RL's Kyrgyz Service have suggested that unspecified officials might be behind the rallies.
The demonstrators blame Kulov for the death of parliamentarian Tynychbek Akmatbaev, who was shot dead on 20 October at the Moldavanovka prison, where he went to help negotiate an end to a prison riot by inmates demanding better living conditions.
Many protesters accuse Kulov of having set up the killing.
At a press conference yesterday, Kulov rejected the accusations and said that the Kyrgyz authorities will not accept any pressure. But he also acknowledged that he would step down if the parliament or president "finds grounds for my resignation," according to RFE/RL.
The protesters in Osh dispersed but vowed to rally again tomorrow.
President Kurmanbek Bakiev has ordered law-enforcement forces to maintain public order.
Interior Minister Murat Sutalinov said after meeting with some of the protesters on 22 October that the Bishkek rally was an expression of a citizen's constitutional rights.
"There was no agreement. We just held talks," Sutalinov told RFE/RL's Kyrgyz Service. "I want to repeat once more: This [rally] is a constitutional rights of citizens to gather and to express their opinions. I don't see any illegal activities there."
Kyrgyzstan was the site of public protests that eventually toppled President Askar Akaev, ushering the opposition to power in what has been dubbed by some a Tulip Revolution.