How does non-electrolytic tinning work?
This is a good question: Tin is above copper in the galvanic series, so logically, copper should displace tin, not the other way around.

Electroless (autolytic) tin plating can be accomplished by disproportionation of a tin compound. $\ce{2Sn(OH)3- -&gt; Sn + Sn(OH)2-6}$, where the less stable Sn(II) yields Sn(IV) + Sn(0), or metallic tin. The divalent tin is usually $\ce{SnCl2}$.

Electroless tin can also be deposited by reduction of a tin compound, e.g.  thiourea reducing $\ce{SnCl2}$ to metallic tin. In a similar fashion, silver nitrate can be reduced to metallic silver to make a mirror on a glass surface. Note that the silver is deposited without  any effect on the glass (other than a nice, shiny coating).