Mariam al-Mahdi — daughter of opposition Umma Party chief and ex-prime minister Sadiq al-Mahdi — was sentenced to a week in prison by an emergency court, Mohamed al-Mahdi, another of the party's leaders, told AFP.
Mariam al-Mahdi and her sister Rabah were among those arrested earlier on Sunday, Presstv Reported.
Protest organizers had called for a march to challenge the state of emergency imposed nationwide by President Omar al-Bashir on February 22.
Bashir's move came after an initial crackdown on demonstrations that have taken place against his rule since December last year failed to rein in the protest movement.
The president has ordered a slew of tough measures to quell the demonstrations, including banning all unauthorized rallies and setting up special emergency courts to probe transgressions.
Sunday's procession was to start at the Umma Party's offices in the capital's twin city of Omdurman.
"As some of our leaders came out of the party office to lead the march, security agents arrested them," said Mohamed al-Mahdi, who is not connected to the former premier's family.
Alongside the two daughters of the former premier, "five other leaders of our party have also been taken away by security agents," Mohamed al-Mahdi said.
"We are still awaiting court verdicts for Rabah and the other five," he added.
He said riot police fired tear gas at the protesters who had gathered outside the party office.
"Police dispersed the protesters before they could stage the march," a witness said.
"Protesters have now launched demonstrations in some areas of Omdurman. Many of them have been arrested," the witness added.
Protests initially broke out on December 19, 2018 after a government decision to triple the price of bread.
The demonstrations escalated into nationwide rallies against Bashir's rule.
Anger has mounted for years over soaring inflation and an acute foreign currency shortage.
Officials say 31 people have died in protest-related violence so far, while Human Rights Watch says the death toll is at least 51, including medics and children.