What I mean is: let's say I've been researching a technology for 20 turns. Then I got one free tech. If I choose to get the one I've been working on for free, are these 20 turns lost, or is it possible to carry the 20 turns' worth resources elsewhere?
Same with production. If I am making a worker for 5 turns and then choose to make a warrior, will I have to start from the beginning or will I have some part of it already done?
There's about three scenarios here, so let's visit them one by one:
1) If you change research projects, your previous research progress remains on the original tech.
2) If you apply a free tech choice to a tech which has research progress, the progress is transferred to the next tech you pick.
3) If you change city production, the hammers remain on the old project for a time. If the inactive project is not restarted within "x" turns, that project will start to lose hammers. "x" depends on game settings and whether the project is a building or a unit (units start decay sooner than buildings).
In some of the previous Civilization games, switching production part way through the process would result in the current production progress being transferred onto the new goal. In Civilization 5 this is not the case.
If you change production in Civilization 5, the current production process on the project you've changed from will stay on that project but eventually begin to decay. More detail about this is covered in Amy B's answer.
What I wanted to add, is if you're building a Wonder and another civilization completes the production of the wonder before you, the current progress of your Wonder construction is calculated as gold and applied to your current gold.
