The eldest son of the murdered Saudi journalist, Jamal Khashoggi, has arrived in the US from Saudi Arabia along with his family.
Salah Khashoggi had previously been barred from leaving Saudi Arabia because of his father's criticism of the country's leadership.
Jamal Khashoggi was murdered in the Saudi consulate in Istanbul three weeks ago, investigators say.
Saudi Arabia's public prosecutor is due to travel to Turkey on Sunday.
Saudi officials deny the involvement of the royal family, including de factor ruler Crown Prince Mohammed bin Salman, in the murder.
But Russia has said the royal family should be believed.
"No-one should have any reasons not to believe them," Kremlin spokesman Dmitry Peskov said.
His comments contrast with doubts among Saudi Arabia's Western allies as Riyadh has changed its version of what happened in the consulate on 2 October several times.
Russia is making clear who it is siding with, says the BBC's Mark Lowen in Istanbul, in another sign of the wide geopolitical repercussions stemming from this murder.
What do we know about Salah Khashoggi?
Two days before arriving in the US, Salah Khashoggi was pictured receiving condolences from Crown Prince Mohammed.
He briefly shook hands with the royal and gave what many see as a cold stare during Tuesday's apparently staged meeting.
Some are suggesting the public display may have been the price for being allowed to leave the country.
A US spokesman said Secretary of State Mike Pompeo had told Saudi leaders that he wanted Salah Khashoggi, a dual US-Saudi citizen, to be able to return to the US and was "pleased" that he had been allowed to do so.
Sources close to the Khashoggis told Reuters that he and his family arrived on a flight from Saudi Arabia and joined his mother and three siblings in Washington.
It is not clear how long Salah Khashoggi plans to stay in the US.
How is the investigation going?
The Saudi public prosecutor, Sheikh Saud al-Mojeb, will be meeting his Turkish counterpart in Istanbul.
The move was announced by Turkish President Recep Tayyip Erdogan in a televised speech.
Turkey also had other evidence which had not yet been disclosed, Mr Erdogan said.
And he called on Saudi Arabia to disclose the location of Jamal Khashoggi's body, as well as the name of an alleged "local co-operator" involved in disposing of it.
The Saudi prosecutor said on Thursday that the killing was premeditated, in a further development of the Saudi narrative.
Saudi officials have also blamed "rogue" agents, and, when Mr Khashoggi first disappeared, maintained he had left the consulate unharmed.
CIA director Gina Haspel is now back in the US - Saudi Arabia's closest Western ally - after a visit to Turkey where she was briefed by investigators. She has briefed President Donald Trump.
No details were released, but media reports said that during the Turkey visit Ms Haspel had heard audio recordings that investigators claim capture the murder.
On Wednesday, Turkey's Sabah newspaper said she had listened to recordings of Khashoggi's interrogation and death, but gave no details about the contents or how the audio had been obtained.
A person "familiar with the audio" told the Washington Post that it was "compelling".
Turkey's Anadolu news agency said on Thursday that 38 members of staff at the Istanbul consulate had been questioned as witnesses.
What has been happening in Saudi Arabia?
State media say suspects are being questioned following information provided by a joint Saudi-Turkish team.
The official Saudi Press Agency reported that on Thursday Crown Prince Mohammed had chaired the first meeting of a committee to reform the state's intelligence services, which was set up following Mr Khashoggi's death.
Last week Saudi Arabia sacked two key advisers to the prince, and arrested 18 people.
On Wednesday Crown Prince Mohammed vowed to punish those responsible for the murder.