As long as you are writing in a style that permits dissonance:
Moving from a perfect fifth to an augmented fourth does not present a problem.
Moving from a perfect fifth to a diminished fifth, "unequal fifths", is also fine.
Moving from a diminished fifth to a perfect fifth is generally frowned upon.
Although enharmonically identical, augmented fourths and diminished fifth are considered functionally distinct; therefore, moving from a perfect fifth to a diminished fourth does not present an issue of parallel motion.
In the P5 -> d5 case, a language clarification is required. What we refer to as "parallel fifths" is really a shorthand for "parallel perfect fifths" (more generally, "parallel perfect intervals"). The latter is prohibited, but P5 -> d5, not being parallel perfect fifths, is okay.
d5 -> P5 is also not parallel perfect fifths, but is generally frowned upon, because diminished intervals should resolve "inward". That is d5 -> M/m3.
Here is an example of unequal fifths. The top voices (soprano and alto) move from a perfect fifth to a diminished fifth. This is perfectly acceptable in 4-part writing. (SOURCE)
When you write in common-practice style, do not use ... unequal fifths: motion from a diminished fifth to a perfect fifth, especially in the soprano-bass pair, since this interferes with proper resolution of the tendency tones (7th scale degree resolving up to the 1st and the 4th scale degree resolving down to the 3rd). (SOURCE)
When the tritone is spelled as a diminished fifth, you, may resolve the 4th scale degree up to the 5th in only one context: when the soprano-bass counterpoint moves upward in parallel tenths. (SOURCE)
Unequal fifths (d5→P5)
• In a three- or four-part texture, a rising d5→ P5 is acceptable ONLY in the progressions I – V4-3 – I6 and I – vii°6 – I6 (no deduction).
• A rising d5→ P5 in other progressions is unacceptable (1 point error).
• The reverse, a rising P5→ d5, is acceptable voice leading (no deduction).
• Unequal fifths in either order, when descending, are acceptable (no deduction). (SOURCE)
Unequal fifths, motion between perfect and diminished fifths is often avoided, with some avoiding only motion one way (diminished to perfect fifth or perfect to diminished fifth) or only if the bass is involved.3 Notice that unequal fifths resemble similar rather than parallel motion, since the perfect fifth is seven semitones and the diminished fifth is six semitones. (SOURCE)
Between the upper two voices it is acceptable to have a perfect fifth move to a diminished fifth in parallel motion. This is called unequal fifths. However, it is not good practice to move from a diminished to a perfect fifth, however.
Diminished fifths ought to “resolve” in contrary stepwise motion into a third. Augmented fourths ought to resolve in contrary stepwise motion out to a sixth. (SOURCE)
Also of interest
This MP&T post may also be of interest: