There's no evidence that they got "out maneuvered". Napoleon ploughed into Russia, sought battle, capture his enemy's capital. His failure wasn't "maneuver" -- it was strategy.

There's a nice bon mot of Talleyrand's, when informed of Napoleon's invasion of Russia, he supposedly remarked "to do what?" That statement, whether correctly attributed or not, gets to the core of the strategic failure of Napoleon's invasion.

Invading a massive nation, with difficult logistics in the best of times, he lost the best part of his army to the environment, not to enemy action. Winter, disease, starvation, the harassment of French supply lines -- that's what destroyed Napoleon's army in Russia. Much of what an army does isn't "maneuver", it's the getting from one place to the next, the horses fed and the men clothed and armed . . . Napoleon pushed 500,000 men into a vast nation in which he did not have the means to accomplish this essential task in the brutal conditions and against competent opposition.

The Russians played their hand competently, but didn't ever win an overwhelming victory in a battle of maneuver; the most important battle of the campaign, Borodino, was a narrow French tactical victory (eg the Russians didn't stop the French from entering Moscow, which was the French objective). Napoleon at Borodino is perhaps less than his best, but he certainly isn't "out-maneuvered. The most significant criticism is that as the Russians began to fall back, Napoleon declined to send in the Imperial Guard to rout the retreating Russians, perhaps losing an opportunity to really destroy the Russian Army, but that's a lot of "what ifs". Similarly at the Battle of Smolensk, you can't say that de Tolly got the better of Napoleon; indeed, that has to be counted as a fairly solid French victory.

The most significant military outperformance by the Russians was in their light cavalry. Napoleon relied on cavalry for forage and for intelligence-- the Russian cavalry was better, and was able to harass Napoleon's army during both invasion and retreat, a steady attrition.

There are many more smaller actions during the retreat, "battles" like Krasnoi and Berezina which are more a continuing skirmish than a battle of maneuver. These actions inflict a heavy toll on Napoleon's already battered army, but it's not the result of any tactical brilliancy, just the steady harassment of freezing men, trying to keep going in the bitter cold. The crossing of the Beresina is an example-- Engineers had to construct a bridge while standing in freezing cold water to enable the Army to escape encirclement . . . they managed to do it, while a rearguard of Swiss troops held off the Russians and permitted the bulk of the Army to escape.

So the best you can say for the Russians is that they were more competently lead by Kutuzov at Borodino than they were by de Tolly at Smolensk or by the Russo-Austrian leaders at Austerlitz, and Napoleon was perhaps slightly less decisive at a key moment than he had been at other times; but that's second guessing. Whatever the case, it wasn't maneuver that defeated Napoleon in Russia.

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Sources:

Dominic Lieven's Russia Against Napoleon is the best secondary source, there are a stack of primary source accounts, of which I particularly like the memoirs of General de Segur and de Caulaincourt. The Diaries of Baron Boris Uxkull give a very good flavor of life in the camp of the army in this campaign.

Charles Minard's celebrated graphical depiction of the dwindling of Napoleon's army in Russia over time and space-- is an inspiration to infographic artist everywhere, a brilliant image that conveys just how the Grande Armee melted away