The video ends with Mr. Shah pointing out New York’s hypocrisy: The city claims to be a “melting pot,” but no one intervened while he was getting harassed. Mr. Shah’s original video, posted on YouTube in June 2016, was a viral hit that attracted more than three million views. A week after he posted it, United Muslims of America copied the video to its group page without the original YouTube link, a process known as ripping. There, Mr. Shah’s video become the Russian page’s most popular post, earning more than 150,000 interactions.

Mr. Shah said when he noticed the ripped video, he wrote to the administrator of the United Muslims account, asking them to add the link to his original YouTube video. His main concern, Mr. Shah said, was that the page was stealing his views. Told that his video had been used by Russian accounts to sow division in the United States, Mr. Shah said there wasn’t anything he could do about it.

“There are always going to be people who manipulate things to their agenda,” he said.

When Being Patriotic posted a brief message last year rallying Americans against proposals to expand refugee settlements in the United States, it was liked, shared or otherwise engaged with by more than 750,000 Facebook users. Eventually, it came across the feed of Len Swanson, 64, a Republican activist from Houston and an avid Trump supporter.

Mr. Swanson, who frequently posts long commentaries on LinkedIn and Facebook, then used the message and photo to open one of his own posts, attacking Hillary Clinton and the Democrats. The message Mr. Swanson borrowed also appears on a conservative meme website, with a photo that at least one newspaper has credited to the United States Navy.

“I usually publish an article several times a week, to keep driving the narrative,” Mr. Swanson said in an interview. He was not bothered, he said, by becoming an unwitting cog in the Russian propaganda machine. “You know we do the same damn thing over there,” Mr. Swanson said. “What do you think — we’re saints?”

In early 2016, Being Patriotic copied and pasted a story from the conspiracy site InfoWars, saying that federal employees had taken “land from private property owners at pennies on the dollar.” The Russian page added some original text: “The nation can’t trust the federal government anymore. What a disgrace!”

Image A Russia-linked Facebook page, Being Patriotic, posted a version of a message rallying Americans against proposals to expand refugee settlements in the United States. The message was reposted by American Facebook users.

This past March, another of the Russian pages, Secured Borders, reposted a video that it attributed to Conservative Tribune, part of the conservative and pro-Trump sites run by Patrick Brown. The video, which falsely claims that Michigan allows Muslim immigrants to collect welfare checks and other benefits for four wives, originated on a YouTube channel called CleanTV. The Facebook post has been removed, but a version remains up on the meme site Me.Me.