As a Khabarovsk local I guess I will be able to make the most accurate explanation. It is going to be pretty long, so that it is easier to understand what's actually going on for a Swede.
Back in 2018 our state held a governor election. No secret that Russian elections are fucked up and rigged, otherwise we would have had Putin stay for so long. Every election in every Russian state is being rigged in favour of Putin's ruling party and its' representatives. People here got so sick of the previous governor, who was a plain retard, but was let to rule because he was loyal to Putin and his regime, and due to that so many people came and vote against him, and all the rigged fake votes did not even make difference, some noname guy won. He won simply because there was no other choice between him and that retarded Putin's puppet.
In fact, a fair and popular candiate will never get an admission to participate in elections, since there is a risk of him actually winning the elections against Putin's candidate. Which means, that noname guy got "pre-approved" by both Putin's regime watchdogs and the FSB (Russian FBI) after being investigated for any possible crime relations. Not to mention that this noname guy used to be a congressman for like 12 years, and MUST have been thoroughly investigated by authorities. He probably was investigated, but nothing suspicious was found.
The guy turned out to be a pretty solid governor, who fought corruption and did a good job developing the local economy. Historically, the Russian Far East has always been in an opposition to Kremlin's authorities, and here in Khabarovsk Putin probably has the lowest electoral rating in Russia, and the rating of the local governor was constantly rising. There was a leaked tape where the new governor was condmned by the Putin's ambassodor of the Far East for having a better rating than Putin.
Right after the 2018 elections we were severely cut with federal funding, and the federal center of the Far East was relocated to Vladivostok (means less fedral funding for Khabarovsk and less funding). And despite that, we still managed to do better economically than ever before. Next year (2019) the Putin's party has got 0 seats in the local state congress (never happened in Russian history before), the situation was getting out of Kremlin's control. The Russian media started to push their propaganda against Khabarovsk governor and congress by distorting the actual economical data.
As you may have heard, we recently had the constitution amendments plebescite in Russia. To sum it up, these amendments basically allowed Putin to stay in power forever and undermined the COE's supremacy over Russian court decisions. The governors in Russia are "supposed" to provide the acceptable results at federal elections for Putin. The poll results in Khabarovsk were among the "worst" in Russia. That probably was the last straw for Putin/government, and the governor was probably ordered to be dealt with.
Now, speaking of the case the governor is being charged, it is full of shit. He is being accused of murdering several people in 2004. According to Russian laws, such cases are considered overdue and are not valid. Moreover, again, he has been a congressman, and was supposed to be investigated. Which means, he either was not investigated properly (then responsible people have to go to prison for that), or these accusations are false. And according to Russian laws, he is supposed to be judged here in Khabarovsk, however, he was for some reason urgently transferred to Moscow.
The new governor used to own a share in a solid metallurgical company. 6 months ago his companion, who owned a 25% share in the company was arrested and transferred to Moscow (he should have been judged here as well). According to the companion's wife, her husband is being tortured badly, as he already sold his share to one of Putin's friends (while being in prison rofl), and the whole case is based upon the companion's testimony against the governor. He probably just gave up to the torments and falsely testified against the governor, as the authorities wanted him to.
Basically, here's what people claim at the rallies:
1. Even if the governor was actually a criminal, then he has to be investigated fairly by the LOCAL court. Once again, that is what Russian laws say. In fact, he was taken to Moscow the same day with the arrest.
2. Should he be proven guilty, solid proofs have to be published.
3. If he is guilty, then how come he has been a congressman for 12 years? People responsible for that admission have to be investigated as well. In fact it looks like he is actually a clean politician (if he was not, some corruption case would have been brought up, not some 16 year old murder case), and just being threatened with life sentence for having a better rating than Putin.
4. There are quite many known criminals in local goverment, who are not being prosecuted just because they are being loyal. Even if the governor is guilty, then investigate everyone, no matter if they are loyal to the regime or not.