Saakashvili left Georgia in 2013 after serving as president for almost a decade, and later was appointed governor of Ukraine's Odessa region. Ukrainian prosecutors accuse him of colluding with Ukrainian businessmen who have ties to Russian intelligence as part of an effort to topple the president.
Saakashvili fell out with Poroshenko over accusations that Kiev was failing to make good on the fight against corruption. But he quit in 2016, complaining that his efforts to root out corruption suffered official obstruction.
On Monday morning, police raided Saakashvili's flat, prompting him to climb onto the roof of his central Kiev building and address supporters below before officers led him down into the vehicle.
Saakashvili called on his supporters to go to Maidan and continue the "peaceful protest".
Ukrainian authorities detained Mikheil Saakashvili at his home on Tuesday morning but hundreds of protesters stopped the police van with Saakashvili inside from driving away.
Saakashvili is credited with pushing through pro-Western reforms in his native Georgia which he led from 2004 to 2013 after rising to power during the so-called Rose Revolution in 2003. He left the country in 2013. He acquired Ukrainian citizenship in May 2015 and was appointed governor of Odessa. Saakashvili forced his way across Ukraine's border with Poland earlier this year, with help from protesters. Hundreds of protesters had been blocking the police van with Saakashvili inside during an hours-long standoff with police.
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"Kiev has surprised us once again and so has Saakashvili", Peskov said.
Speaking at a news briefing in Kyiv on Tuesday, Lutsenko said Ukrainian law-enforcement agencies suspect Saakashvili of aiding and abetting members of criminal organizations and covering up their activities (Ukrainian Criminal Code Article 256), which would justify requesting his round-the-clock house arrest.
Saakashvili spearheaded several protests in Kiev, but they typically drew fewer than 4,000 people.
In September, the Georgian Chief Prosecutor's Office asked Kiev to take measures for Saakashvili's extradition.
Analysts in Kiev, however, don't see the Saakashvili case sparking protests big enough to challenge Poroshenko.