Litvinenko posited this to be the case (Russian influence with ME terrorism) nearly a decade ago. Things didn't end up too well for him, obviously. (In fact, openly mocking good sense and international outrage, one of his killers now serves as a Russian MP. One of those "Only in Russia" kind of things.) As others, including Schindler, have written, there's evidently still a significant debate about al-Zawahiri's and other al-Qaeda bigwigs mysteriously baffling time in southern Russia and the North Caucasus back during the 90s. And these questions aren't just about whether or not it was to serve the Chechen cause.
By allowing and promoting the emigration of Islamists, Russian authorities achieve several aims. They clear the volatile North Caucasus of Islamists and reduce violence in the country. The export of militants to the Middle East also serves Russian foreign policy goals, most principally to cause as much volatility in the Middle East as possible in order to drive up oil prices. High oil prices are practically the only way for the current Russian regime to survive in the long run. What appears to be a widespread practice in Russia, agent agreements signed between the Russian security services and Islamist recruits amounts to the Russian state recruiting militants for militant organizations in the Middle East. This policy of creating problems for neighboring countries, while having an Islamic underground movement on Russias own territory, is likely to backfire sooner or later. Even though Russia temporarily solves its problem in the North Caucasus and creates headaches for others in the Middle East, this strategy is unlikely to lead to a long-term solution for either of these regions.
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I would add the most likely motives, if true, would have to also include a desire to further destabilize US foreign policy goals.
Bashar al-Assad has to be sitting in his Damascus palace right now wondering what in the hell he did wrong and why Russia now hates him so much. Well, Bashey Boy, you're pretty expendable and some things are more important than keeping up the dictatorship as far as the Russians are concerned.
Lastly, I'd like to add that, given Kadyrov's trending towards open hostility with the Kremlin and now this revelation, my official Volprof's Countdown to the Third Chechen War clock is moving closer and closer to midnight. As the US well knows (now, I hope at least), playing with fire can very well backfire and, indeed, often does.