I think there are actually two separate (though interrelated) issues that you're asking about here. On the one hand, there's a question about music with pitches tuned to the overtone series of a particular fundamental ("should I base the tuning on the same tonal center?"), and on the other hand there's tuning based on simple ratios between integers ("how should an F major chord be tuned?"). I think your question is ultimately more about the second issue, but the first is significant as well.
If I am tuning something entirely based on the overtone series above C, then the I chord (c-major triad) is pretty clear-cut: I would use the C, E and G that appear in the first five partials and will thus have the standard just major triad relationship 6:5:4. This means that the E will have a frequency that is 5/4ths of the C (5:4 is the standard just major third) and the G will have a frequency that is 6/5ths of the E (6:5 is the standard just minor third). Automatically, this also means that the G will be 6/4ths (= 3/2nds) of the C (3:2 is the standard just perfect 5th). Even something as simple as your example of moving to the subdominant chord, however, is rife with complexity. For one thing, the first pitch in the overtone series that is close to an F is the eleventh partial, but that's nearly a quarter-tone higher than the F that is a fourth above (= a fifth below) my C fundamental. Although it might be cool to play around with this kind of subdominant, it definitely isn't what is usually meant by the term. The next partial that is a candidate for F is the 21st partial. It's definitely closer to the expected pitch, but still not very close, plus these higher partials are rarely what people are talking about when they talk about trying to use just intonation for better tuning.
This is where thinking in terms of just ratios rather than a specific fundamental makes far more sense. We almost certainly want the root of our F major triad to be a just perfect fourth or fifth away from the C, which means it needs to be 4/3rds or 2/3rds the frequency of the C. If we went a pure just-intonation major triad, then we just add an A and a C that are in the 6:5:4 relationship discussed above. Happily, this gives us the same C pitch (give or take an octave) as our original fundamental; these kinds of same-note interrelationships are common with the basic standard harmonies of a major key, but all kinds of complications ensue when more complex relations are explored. Theoretically, we could say that the F major triad is being tuned according to the "key" of F major, but that would miss the point: the 6:5:4 triad on F is absolutely part of the key of C major even if its notes aren't present in the earlier partials of the overtone series of C.
From a practical performance perspective, then, the issue tends to just revolve around 6:5:4 major triads and (often) 15:12:10 minor triads. Those ratios are entirely relative of course. For the most part, this means that the thirds of major triads need to be a bit flatter than equal temperament would dictate, and the thirds of the minor triads need to be a bit sharper than equal temperament would dictate. If my tonic were 100 Hz, then my F would be about 133 Hz (roughly 4/3rds of the tonic). My A would be about 166 Hz (roughly 5/4ths of the F), and my C would be 200 Hz (roughly 6:5ths of the A, or twice the original tonic). The reality is both vastly more complex, but also vastly more intuitive, and I would strongly advise against being overly dogmatic about any of it. Context can, and often does, override such slavishly local details about idealized relationships.