I'd want to precizise the answer of Laurence:
I think the point of MOVEABLE Do is that it CAN move! (But it must not - even if it theoretically does!) It is depending of the individual taste whether we want to relate the syllables to "modulation"in each phrase or only in longer sections:
e.g.
In a half-cadence we're ending on the dominant: The melody in C is ending on a G-major chord. Now we can name the leading tone to the root note (F#->G) ti->do. G becomes the new root tone (only for 1-2 measures) or we don't move the Do and remain in C even there is a short modulation to the dominant key and name F#->G as Fe->Sol.
So this depends from your individual taste and personal goal and also from the specific situation in a piece:
I prefer sometimes to keep the Do in the tonic of a piece in purpose not getting lost in the harmony.
Another example are the ascending fifths.
C-D-E-F -> D-E-F#-G -> E-F#-G#-A
you can perform this pattern 3 times as So-La-Ti-Do, but to me it makes more sense to interprete them as what they are: doremifa, remifesol, mefesela.
When I was a beginner in solfege I even replaced the upper tetrachord of the melodic minor scale mifisila by the 2nd tetrachord of the major scale:
la ti do re mi mi-> sol: sol la ti do.
There are now rules or laws. Your personal prferences decide about the labelling: This may depend of ear training, simplicity, clarity, modulation training, harmonic correctness etc.
Edit:
I was using the movable Do even when I had to sing 5 Lieder by Webern when studying solfege! This may explain the best what movable do can be:
A help for ear training (s. my Example of Webern!) - or using for analysis. In the second case it should be accurate - as possible - with the harmonic progression.