Without getting into a debate on the type of carbonation, some people (including myself) find natural carbonation different than/preferred over forced carbonation.
In this case, you'd transfer the contents of the carboy over to a keg then just seal her up. Ideally, you'd know the expected FG (via experience with this recipe or a fast ferment test), so you know what gravity would be "too late" to make the transfer, and thus undercarbonating it (naturally). You'd want to err on the side of making the transfer with higher gravity than your target, as that's what the spunding valve does - acts as a pressure relief valve when the pressure gets above your target.
Adding new yeast isn't necessary, even if you miss your window to transfer (and run the risk of undercarbing), because there's still plenty of yeast in solution. In that case you'd either add sugar (if you were looking to naturally carbonate - like one big bottle), or you'd force carb with CO2 the rest of the way. You shouldn't, however, assume that the remaining fermentation in the keg is going to scrub all oxygen picked up in the transfer, so you should still practice good anti-oxidation measures (ie. purging the keg with CO2 before transferring).