Some say they were attacked and harassed on their way to the airport. Others claim they had their passports damaged by Russian border officials.
They were all on their way to New York to attend a United Nations World Conference on Indigenous Peoples, which began on September 22.
Five delegates to the conference claimed separately that they were prevented from traveling -- either by unknown assailants or by border guards.
Nadir Bekirov, a Crimean Tatar activist, says he was attacked by a group of masked men while en route from Simferopol to Dzhankoy, in Russian-annexed Crimea on September 18. He says masked men stopped his taxi, pulled him out of the vehicle, and took away his passport by force.
Bekirov, who heads the Foundation for Research and Support of Indigenous Peoples of Crimea, said he planned to take a train from Dzhankoy to Kyiv, and then fly to New York. But without a passport, he was unable to leave Crimea.
Russian media reported that another Crimean Tatar member of the delegation, Gayana Yuksel, had her passport confiscated by police officers on her way from Dzhankoy to Kyiv.
Meanwhile, in Moscow on September 18, Rodion Sulyandziga, the head of the Russian Center for Support of Indigenous Peoples of the North, said border guards seized his passport at the capital's Sheremetyevo International Airport.
Sulyandziga said the officers told him a page was missing on the document.
A day later, on September 19, Anna Naikanchina, from the same organization, had a similar experience.
Russian media reported that officers took her passport at the airport, and that when they returned the document, Naikanchina found out that several pages of the passport were missing.
Valentina Sovkina, the speaker of the Saami parliament of the Kola Peninsula, made it to New York for the conference -- but only after several incidents on the way.
Sovkina said the tires on her car were slashed when she was about to leave her home in Murmansk for the airport on September 20. She managed to repair the damage, but was twice stopped and questioned by police on the way.
Sovkina posted on her Facebook account photos of traffic-police cars parked alongside a road, and police officers who appear to be checking documents and inspecting a vehicle. She wrote that the police asked her if she was delivering weapons to Ukraine.
Sovkina said the questioning caused her to miss her flight. She ended up flying to Norway and taking a connecting flight from there to New York.
Most Outspoken Critics
Russian authorities have not publicly commented in the incidents.
Russia has lodged a formal protest with the United Nations over the participation of Crimean Tatar leaders Mustafa Dzhemilev and Refat Chubarov at the conference.
Dzhemilev, a rights activist and Ukrainian lawmaker, and Chubarov, head of the Crimean Tatar Mejlis assembly, have been vocal critics of Moscow's annexation of Crimea.
The two-day conference was attended by more than 1,000 delegates from indigenous communities, as well as experts and high-ranking officials.
According to media reports, only 23 members of the 28-member delegation from Russia managed to attend.
Pavel Chikov, the head of the Kazan-based Agora rights group, told RFE/RL's Russian Service that the Kremlin was trying to deprive indigenous groups of the opportunity to directly communicate with the outside world. He noted that those who were singled out by the authorities were among the most outspoken critics of the Moscow authorities.
Chikov said the crisis in Ukraine had made the Kremlin sensitive to any "mention of federalization, self-determination" or "rights of indigenous people" in Russia, adding that law enforcement and security officials "have a carte-blanche to pressure" the activists.
Russia is home to some 180 ethnic groups whose populations vary from several hundred to millions.