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This seems to be a mostly false tale, though it is based on evidenced reports.

From the archives of the French Police, in a memo sent on 15 June 1810 to Savary, the newly-appointed Minister of Police for 15th June 1810.

15 - Boulogne. Escaped Englishmen. The customs officers arrested near Boulogne the names Mogg, Lewis, Partson, Altsford, English prisoners of war who had escaped from Arras ; they intended to embark on a skiff that they had managed to build with canvas and ropes that they had brought with them, and wood that they had cut in the forest where they had hidden ; they had made this boat waterproof by means of a thick layer of tallow. These prisoners, from Arras to the wood in which they had hidden near the coast, had walked only at night, using the moon and the stars to guide them. The administration of the Navy tested the canoe made by these four prisoners; 6 men embarked, steered it with oars and held the sea without a drop of water entering. This invention was generally praised. These prisoners told the general commissioner who questioned them: "If the Emperor knew the boldness of our undertaking, he would grant us our freedom".

In the archives of Napoleons correspondence however, we can find that the imperial family stayed in Boulogne at the end of May 1810, leaving on the 26th, but Napoleon did not set foot there after that, as he was in Saint-Cloud and Rambouillet doing governing stuff for the rest of the year. He did however correspond regularly with the local magistrate Savary.

It is possible that Napoleon read the plea of Mogg and his companions, and it is imaginable that he ordered that they be set free, but there's no mention of that in his letters to Savary, despite an amazing display of micro-management. There's no way Napoleon took a couple of days to go to Boulogne and talk to the sailor in person though.

Most probably, what happened is that the true story of the daring attempt of the four sailors, and their remarkable plea to Napoleon, may have been famous in Boulogne, and that it somehow morphed into a much better "Napoleon freed a British sailor, gave him money, and sent him home" story with its fairy-tale ending, and that it was this story that circulated both locally and among British sailors, and was reported to Las Cases by his sailor acquaintance. Las Cases says that Napoleon confirmed the stories but, as we saw, some are indeed true.

There are, however, real tales of Napoleon's generosity towards British sailors, and I'll conclude this answer with the following anecdote told by Lewis (1962), which took place a few months after the Mogg story.

On December 27, 1810, the Indian country-ship Elizabeth, Captain Robert Eastwick, drifted on to a shoal off Dunkirk in a terrific gale. Of a total complement of 380, only twenty-two, including Eastwick, braved the appalling surf, and reached the shore more dead than alive. They were instantly carted off to the town gaol and locked, white officers and lascar seamen together, in a foul cachot. But the Dunkirkers, who were known to have secret sympathies with the British, were, openly for once, furious. Having watched with their own eyes the survivors’ long and desperate fight for life, and having spontaneously cared for them and revived them with the most solicitous kindness, they were scandalized that they should thus be hauled off to a dungeon: and their rage and shame were further increased when it came out that one of the survivors had been foully murdered. Edward Tench, a lieutenant in a Ceylon regiment, was a passenger. He was not with Eastwick in the boat, but had somehow managed to swim ashore alone. There, exhausted, he was found by a soldier and a customs officer, who promptly dispatched him for the sake of the money he carried. This was too much. With one voice they appealed to Caesar, the commandant himself leading.

The Emperor, just then, was in the throes of his argument with the British Government; squirming, wriggling, cheating right and left, and loudly proclaiming in public that he would not give way an inch to the utterly unreasonable demands of the enemy. Not a single Englishman would be exchanged until their rulers mended their vile ways. In fact, the Moniteur had only just printed a message from the imperial pen declaring as much. Add to this the well-known fact that His Majesty did not like Dunkirkers, who, as he knew very well, did not like him. Save perhaps for the Bretons, they were the last people whom he would willingly oblige. None the less — and to everyone’s surprise, including Eastwick’s — he sent for the petition as soon as he heard of its arrival, read carefully the account of the shipwreck, and, with that same hand which had written so recently to the Moniteur, signed an order for the immediate and unconditional release of the whole party, the French Government to bear all expenses. He did not demand, or even hint at, any exchange. But of course he got one. Before he left for home, the grateful and far-sighted Eastwick held a consultation with the leading Dunkirkers, who presented him with a list of 150 of their townsmen who were prisoners in England; and the first thing he did on reaching London was to visit the Transport Board, where he told his story. Instantly twenty-two Dunkirkers departed for France, and they were the ones picked by their own folk as being the most deserving of release.

original, sources and text I am mostly repeating here | r/AskHistorians Reddit

Well, yeah, and everybody clapped