In previous Pokemon games, titles throughout the generation all took place in the same region(s). However, since FireRed and LeafGreen are remakes, they do not take place in Hoenn as Ruby, Sapphire, and Emerald did, and thus do not share the same regional Pokedex. This and a few other factors mean connecting with them is a bit more complicated than usual. Once you collect the Ruby and Sapphire items in the Sevii Islands and give them to Celio, all restrictions the game places are lifted.

Union Room

Although you cannot use the Wireless Adapter with Ruby and Sapphire, Pokemon Emerald was programmed to take advantage of it. However, you cannot battle or trade Pokemon within the Union Room until the Ruby and Sapphire are collected.
Battles

This requires giving Celio the Ruby and Sapphire. For the first time there are a few moves that can only be learnt in specific versions, these versions being FireRed and LeafGreen. Such moves can be used in battle without a problem. As well, all Pokemon (even those not found in Ruby and Sapphire) can be used.

In the case of Deoxys, its Attack and Defense forms remain exclusive to FireRed and LeafGreen respectively. Deoxys always carries the stats of its native version's form (Normal in Ruby/Sapphire, Attack in FireRed, Defense in LeafGreen, Speed in Emerald) Despite this, Deoxys will graphically appear as the form from whichever version you're playing.

Trades

This requires giving Celio the Ruby and Sapphire. Any Pokemon not found in the Hoenn Dex will unlock the National Dex if traded to Ruby and Sapphire. As well, Deoxys will change its form automatically whenever it is traded between the different versions. Moves not normally learnt in Hoenn, like Blast Burn, are freely tradeable
Future Generations

A few minor connectability quirks concerning future generations; when you play Diamond, Pearl, or Platinum with these games in the GBA slot of the Nintendo DS, certain Pokemon will appear in locations after you acquire the National Dex. Pokemon brought up from these remakes as well will list their native region as Kanto, instead of Hoenn. As one final note, Pokemon from FireRed and LeafGreen cannot be used to collect the Time Travel Award in Pokemon Omega Ruby and Alpha Sapphire.