In the case of Generation 1 to Generation 2, you cannot transfer items from your bag to Generation 2, but the Pokemon you obtain may hold an item when transferred. The item you get from transferring is dependent on the Pokemon's catch rate for that game. This page on Bulbapedia shows a list of Pokemon by their held item in Generation 2. The items are generated in Generation 2, and sending held items back to Generation 1 is impossible.
In Generation 4, Pal Park allowed held items to be transferred on Pokemon from Generation 3. Just about every held item can be transferred, except for mail (the Pokemon won't be allowed to be transferred) and the e-Reader berries (which will become Enigma Berries).
Every other generation has your items returned to your bag. Poke Transporter doesn't include any held items on Generation 1 Pokemon (probably since a handful were Gen 2 exclusive, such as the Brick Piece), and held items on Generation 2 Pokemon get sent back to the bag when transferred.
So, in general, no, you can't directly transfer items except from Generation 3 to Generation 4, though you can potentially influence it in Generation 2 by capturing certain Pokemon. In every other generation, it's impossible.