A former steelworker-turned-Communist apparatchik, Nazarbayev ruled the oil-rich Central Asian nation from 1989 until his resignation in March, when he installed a loyal successor while retaining sweeping powers for himself.
Nazarbayev's new title is Yelbasy, or national leader, and he is also head of the influential security council.
His July 6 birthday is a public holiday officially linked to the inauguration of the former Soviet republic's capital, which was renamed Nur-Sultan after his resignation.
Protesters gathering near a square in Nur-Sultan on Saturday were met by hundreds of police officers in riot gear who quickly dispersed the rally, detaining dozens of people and taking them away in buses, said a Reuters correspondent at the scene.
In Kazakhstan's biggest city, Almaty, hundreds of riot police awaited protesters gathered near the main football arena.
A Reuters correspondent at the scene saw them detain dozens of people, some of whom shouted "Wake up Kazakhstan!" and "Old man, go away!," a commonly used anti-Nazarbayev slogan.
Police also detained a handful of people each in Almaty and Nur-Sultan on the eve of the protests, which were organized by supporters of Mukhtar Ablyazov, a fugitive former banker and government minister who lives in France.
Kazakhstan's Interior Ministry said in a statement police had acted "to prevent disturbances of public order." Public protests are illegal in Kazakhstan unless they are agreed in advance with the authorities, who rarely allow them.Although they might have been a nuisance to the government, the protests were smaller than those which rocked Kazakhstan's major cities last month after the snap presidential election which Nazarbayev ally Kassym-Jomart Tokayev won with about 70 percent of the vote.