Do Ticket to Ride expansions work with the Europe set?
No, you only need one of the base games (Europe or USA) in order to access the train car cards, scoring markers and the player carriage pieces. It doesn't matter which base game you own. For example, the rules for Heart of Africa tell you:


  This game is an expansion and requires that you use the following game
  parts from one of the previous versions of Ticket to Ride:
  
  A reserve of 45 Trains per player and matching Scoring Markers taken from any of the following:
  
  
  Ticket to Ride / Ticket to Ride Europe 
  
  
  110 Train Car Cards taken from:
  
  
  Ticket to Ride / Ticket to Ride Europe / USA 1910 expansion
  


You can also confirm this yourself by looking at the Games of Wonder website descriptions for each map, for example Asia. They all say the same thing:


  NOT A STAND-ALONE GAME! Requires an original copy of Ticket to Ride® or Ticket to Ride® Europe to play.