The flux is at least as important as the alloy. Half the reason Miti's solder is so nice to work with is the Sn100C alloy, which is generally considered "best of breed" for lead free alloys. (It makes those nice shiny joints we all love.) But the other half is that it's loaded with 3% of powerful flux: FCT's NC601 is a potent "RA" "Rosin Activated" type flux. In general "RMA" (Rosin Mildly Activated) is standard for fluxes, RA is one step up. Often the manufacturer will say you should clean RA flux residue off, but it's rare for that to be actually important. "No-Clean" fluxes are often impossible to clean off when you do need to do it, which can be annoying. This is getting better with more recent fluxes, or maybe the marketers are just labeling more things as no-clean.
The other thing you should be aware of is that mixing solder types is very bad for joint strength. Getting a bit of lead in a lead-free alloy can really damage its mechanical properties. So if you are reworking leaded joints with lead-free, you need to get as much of the old solder out as possible. Going the other way is similar but a bit more forgiving. I have seen reworked joints where this was not done put under mechanical stress... and they split halfway down the middle of the joint, at the boundary between the two types of solder, not heated enough to mix. Oops.
It's best, if you know what the old solder was, to just use more of the same.