The demonstrations were the first since the ruling military council and civilian opposition agreed in principle to a power-sharing arrangement ahead of elections. The deal has yet to be finalized and signed.
A meeting between the two sides planned for Saturday was postponed to Sunday, a leader of the Forces for Freedom and Change (FFC) coalition said. But the military council denied the meeting was being delayed.
"Saturday's session will discuss the constitutional document as determined by the mediation," state news agency SUNA said, citing the council.
African Union mediator Mohamed Hassan Lebatt had said Thursday the council and the FFC would meet Saturday to study and ratify a constitutional declaration. They had agreed to a political declaration that determines the transition's different institutions, he said.
After the meeting, the Sudanese Professionals' Association, which leads the FFC, said, "The draft constitutional declaration is 'not final' and is not open to final signature in its current form."
The constitutional declaration's signing was pushed to Sunday for further consultations based on FFC's wishes, Lebatt said on Sky News Arabia earlier Saturday.
In Khartoum Saturday, thousands protested on Sitteen Street, a major thoroughfare in the capital, a Reuters witness said. Some lit candles to remember those killed at the protest camp on June 3, while others lit the torches on their mobile phones.
"We came out to express our opinion and convey our voice and salute the memory of our eternal martyrs," said protester Mostafa Sayed Ahmed.
Six vehicles belonging to the paramilitary Rapid Support Forces (RSF), each carrying about six men armed with assault rifles and sticks, drove through a portion of Sitteen Street as protesters chanted "Civilian!" at them, a Reuters witness said.
A witness also saw more than 20 RSF vehicles carrying men in riot gear at Abu Janzir Square in the heart of Khartoum.
"Look at these crushed people," said Hussein Ismail, a middle-aged demonstrator who was chanting "We either get their rights or die like them!"
"Their demands are clear, which are a civilian government, a democratic state, which is a people that call for justice and peace and love."
Several hundred also demonstrated in Khartoum's Burri neighborhood, a working-class district and the cradle of many of the protests. RSF troops stood on roads surrounding Burri, armed with sticks.
"Blood for blood, even if (we get) civilian rule!" protesters chanted.
Security forces used barbed wire to block the main road leading to the Defence Ministry compound, the site of the protest camp crushed by security forces in June. At least 128 people were killed during the raid and in the two weeks that followed, according to doctors linked to the opposition.