As a critic of the Kremlin Babchenko stood in unofficial elections organised by the opposition in 2012.
Babchenko sparked a backlash in Russia for his comments in a 2016 Facebook post on a Russian military plane crash.
Vyacheslav Volodin, speaker of the State Duma, told Russian news agencies Wednesday that Russia would be happy to help with the investigation if Ukrainian authorities requested it.
Ukrainian Prime Minister Volodymyr Groysman said in a social media posting late on Tuesday he was convinced what he called "the Russian totalitarian machine" had not forgiven Babchenko for what Groysman called his honesty. The wounded Babchenko was found by his wife and died in an ambulance on the way to the hospital.
Mr Gerashchenko said Mr Babchenko's killer was waiting for him on the staircase inside the journalist's building and shot him in the back as he was going out to buy bread.
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Two years ago, Pavel Sheremet, a Belarussian journalist known for his criticism of his home country's leadership and his friendship with the slain Russian opposition leader Boris Nemtsov, was blown up in a vehicle bomb in central Kyiv.
Leaders in Moscow, conversely, pinned the blame on Ukrainian authorities, saying that the government was unable to adequately protect journalists.
In March 2017, renegade Russian lawmaker Denis Voronenkov was shot and killed at the entrance of an upscale hotel in Kyiv.
So it seemed like Russian business as usual in the murder of Arkady Babchenko, who reported from Ukraine and offered strongly critical commentary about Putin and his regime. He also published several books based on his wartime experiences. He left Russian Federation in February 2017, saying he was receiving threats and concerned he might be jailed. "Now, for the most part, the people that Russians hate-it's all of you".
He moved to Kyiv last fall, where he worked as a host for the Crimean Tatar TV station, ATR.