I've not seen any sources on this, but here's an explanation based on the viewpoints of several people I know.

The shift you've noticed stems from the fact that people dislike being preached to. These people you're referring to originally held relatively pro-Russia views (the reasons for this are varied and could easily be a separate question, if you want to know them). The general feeling is not that the war is justified, but rather that it is understandable, and the US is the ultimate cause of the war. Sort of like if you kick a dog several times and the dog bites you, you could argue that the dog is at fault, or you could say that you provoked it. 

Western media has consistently tried to push the viewpoint that the dog is at fault. Since they never acknowledge that the dog has been kicked for decades, or that the West is the one that did the kicking, or the irony of criticizing the invasion when they also invaded Iraq looking for non-existent WMDs + caused regime change (and later turmoil) in Libya/Afghanistan etc., or that [Russia should accept hostile bases on their borders even though they themselves won't][1], this is not very persuasive. 

For the same reason people downvote push questions here on Politics SE, people also get turned off by the constant moralizing coming from Western media.

Note this doesn't mean people support the war. More likely they hate the war, because war is bad and causes suffering. It also doesn't mean people hate Ukraine and want them to lose.<sup>1</sup> Ukraine is just the unfortunate party caught in the middle. But it does mean they're less likely to support sanctions on Russia.

<sup>1</sup>A minority of people might want Ukraine to lose the war just to give the media the middle finger. You can see an analogous feeling among Republican voters in the last two US elections, when the media seemed to be overwhelmingly pro-Democrat. 

**Edit**:

>I understand Russia's sound strategic reasons to dominate Ukraine. And I get that some people dislike the West and root for Russia (and anyone else who stands up to them). But at the moment of invasion, when one sovereign nation invades a smaller one, ostensibly unprovoked (and I don't count anti-Russian language laws, foolish though they may have been, as justifying an invasion), how can an individual who has reasonably good access to news (i.e., doesn't live in, e.g., China) think to himself "yes, Russia was right to invade, I hope they win"?

This asks about actually supporting the war, which goes beyond the relatively neutral position described in my original answer. In this case, check out this old [tweet][2] by Sergey Karjakin. Karjakin is a chess Grandmaster born in Crimea who swapped to Russia in the 2000s, so he has a front-seat view of Ukraine-Russia relations. The tweet itself references [the following Telegram post][3], which says (machine translated): 

>Are there people who like war? I suppose if there are, they are a minority. And the least lovers of war are among generals and officers. Because they know very well what the price of victory is.
>
>There is a lot of noise on social media now that the war in Ukraine must end. But that's exactly what our polite people are doing now. For those who have a bad memory, I can remind you about the coercion of Georgia to peace. Until 08.08.2008 - constant provocations and shooting on the border, loss of life. After that, there is peace, silence and tranquility. And no one in Georgia even thinks of trying to attack South Ossetia and Abkhazia, let alone Russia.
>
>We did not start the war in Ukraine, and it began a long time ago - with the first killed not even in Donbass, but on the streets of Kiev in January 2014. But now we have an opportunity to end this protracted war.
>
>Therefore, I am extremely surprised to see those who are now calling on Russia to stop. Those who are really for peace, let them send words of support to their friends and relatives in Ukraine. The long-term plague will soon subside. Artificial enmity will cease.
>
>Those who take to the streets of Russia with Ukrainian flags are simply deceiving themselves. The world is closer now than ever before. In addition, the risk of provocations is high, because the real enemies of peaceful Ukraine now really need videos of "aggression". This must be understood first of all.
>
>I am for peace and tranquility on the Ukrainian land. And since the people of Ukraine themselves have not coped, it is our sacred duty to help them. Our soldiers came in peace. Any support for the Kyiv regime is support for the war.

Interpret as you will.

  [1]: https://politics.stackexchange.com/questions/72830/why-does-australia-not-tolerate-china-building-a-base-in-solomon-islands-if-they
  [2]: https://twitter.com/SergeyKaryakin/status/1496888874303467523
  [3]: https://t.me/annashafran/5708