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 finalised 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 told Reuters. 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 on Thursday the council and the FFC would meet on 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 on Saturday.
In Khartoum on 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.
"The fate of the former regime, to us, until now, is vague and many things are unclear," said Osama Iskandar, a young protester, referring to the government of Omar al-Bashir, who was overthrown by the military on April 11.
"We will be on the Sudanese street level until our demands are fulfilled," he said.
A Reuters 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-age 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 calls for justice and peace and love."
Several hundred also demonstrated in Khartoum's Burri neighbourhood, 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 a main road leading to the Defence Ministry compound, the site of the protest camp crushed by security forces in June, a Reuters witness said.
At least 128 people were killed during the raid and in the two weeks that followed, according to doctors linked to the opposition. The government confirmed at least 61 deaths.
Across the Blue Nile, hundreds protested in the neighbourhoods of Shambat and al-Mazad in Khartoum North.
In Khartoum's twin city of Omdurman, hundreds demonstrated on al-Arbaeen Street, a major artery. Thousands also turned out in Wad Madani, capital of Jazeera state, while others protested in Port Sudan, capital of Red Sea state, and Al-Ubayyid, capital of North Kordofan.
General Mohamed Hamdan Dagalo, deputy head of the Transitional Military Council and head of the RSF, which controls Khartoum and whose members are accused of violently dispersing the sit-in outside the Defence Ministry, defended the latter's role in maintaining security.
"Rapid Support are not angels, but we prosecute every offender," Dagalo, known by his nickname Hemedti, said in a televised speech. "Were it not for Rapid Support, Khartoum's situation would have been different."