Do destination cards have to be completed in a particular direction?
All you need is an unbroken chain between the two cities. It doesn't matter which way you started it, how many cities are in between, or whether it is the most direct route.

From the rules:


  Each Destination Ticket includes the name of two cities on the map and
  a Point Value. If a player successfully completes a series of routes
  that connect the two cities, they will add the amount of points
  indicated on the Destination Ticket to their point totals at the end
  of the game. If they do not successfully connect the two cities, they
  deduct the amount of points indicated.


The rules for Ticket to Ride: Europe are similar.


  If, by the end of the game, a player has created a continuous path of his color plastic trains between the two cities named on a Destination Ticket he holds, he scores the additional points indicated by the Point Value on the Ticket. If he has failed to complete a continuous path between those cities, he deducts the Point Value on the Ticket from his total score.