I think you need to be set straight about the entire process.
Usually, a company negotiates with the company that originally produced the series to acquire the rights to produce an English version. This agreement is called a license, which gives the licensee access to the original Japanese masters (usually Beta or DigiBeta transfers, but sometimes electronic storage for newer series, or even the original Telecine film reels for older shows) and the rights to produce, market, and release to the public its own version of the product. A company doesn't buy pre-dubbed or pre-edited episodes, it translates (or doesn't, in 4Kids' case) and dubs them itself (or sublicenses the dubbing out to a recording studio to do it for them). Also, a license generally comes with a built-in expiration date, usually a certain number of years from the date it goes into effect.
In the case of 4Kids, the company purchased the rights to produce an English version from Toei Animation, then subsequently made an English version that was heaviliy censored and rewritten from the original version. This version (unsurprisingly) wasn't as popular as 4Kids had hoped, and their license was not making them as much money as they wanted. Now, either 4Kids did not renew its license, or they sold it off, allowing FUNimation to obtain the rights to produce a completely different version. This new version will be produced in-house by FUNimation, based once again on the original Japanese masters, but hewing much more closely to the original format.
If you have any further questions, I might be able to answer them.
No, no, I know the stuff of the License, but that's the problem, I tought that in this case, if the company that is airing One Piece where I live actually bougth the rights to broadcast the "edited" episodes of 4kids (Even Alh Khan names appears on the credits.), then could it be possible that they continue airing One Piece? I mean, could there be a problem if they want to get the rights to air the "already edited" episodes of Funimation in Mexico? Or would it be cheaper for them to get the "uncut" episodes from the current distributor of One Piece, funimation?
In short, I'm just wondering if they would have to "wait a while" before being able to import the Edited Episodes of Funimation…
Another note: One Piece do air in CN where I live (And in a Public Channel where the original opening and ending was included, but it was still the 4kids episodes), would that make things even more simple, even tough the channel has a completely different managament? (I think it's kinda like a Monopoly, dunno if I'm right..)