Discarding a Card to Go for a Walk

• When a player decides to go for a Walk, they discard 1 card from their hand to the face-up discard pile.
  This is stated as:

  "A player does this by discarding 1 card from their hand to the face-up discard pile, taking the top face-down card (without looking at it) from either the Tokyo or Kyoto deck, and placing it face-down in their itinerary as they would any other Activity card."
  (page 7)

• The discard happens immediately as part of the process of going for a Walk.

Using a Card Just Drawn via Research

• Players may use Research tokens to draw additional cards after their regular draw for the round, but before passing cards at the end of the round.

  "At any time after drawing their regular cards for the round but before passing cards at the end of the round, a player may discard a Research token to the supply to draw 3 cards in any combination from the Tokyo and Kyoto decks to their hand. The player must then IMMEDIATELY discard any 3 cards to the face-up discard pile in the center play area."
  (page 5)

• There is no restriction in the rules that prevents a player from using a card they just drew via Research to go for a Walk. The only requirement is that the card is in their hand.

Summary

• The card discarded to go for a Walk is placed in the face-up discard pile immediately.
• A player may choose to go for a Walk with a card they just drew via Research that turn, as long as it is in their hand and before passing cards.

Source: pages 5 and 7.

Source: Base Rules (pp. 3, 5, 6, 7, 8)