Not all notes are equally important; choosing which stack of notes to be labeled with a chord symbol should reflect this!
The measures quoted is from the beginning of Chopin's Nocturne in A Flat Major Op. 32 No. 2. Hear this sample performance where you can clearly discern that the top notes are melody and the first two measure prominently and ominously announce a motif to acclimate the audience of what is to come.
Selecting the most important notes, for the 1st measure I would choose the chords on the 1st, 3rd, and 4th beats to be labelled. But I would ignore the top B-flat in the second half of the fourth beat, which is simply a melodic passing tone, so no chord label there.
In contrast, the most important moment in that measure is the surprise transition from Db to Dbm as pianist Grigory Sokolov highlights the F-flat in his performance. Thus, for analysis-sake, the F-flat is NOT merely a passing tone but a critical note in the chord transition from the 3rd beat to the 4th beat, making the 4th beat chord a MUST to be notated also, even though it's an upbeat. In Claudio Arrau's performance the semitone descent from F to F-flat to E-flat are more evenly rendered (with a decrescendo) than Sokolov's interpretation, giving us a clue that those 3 notes are the most important notes of the 3 chords and that only those 3 chords should have chord symbols.
When the first section starts, every beat is significant to have a chord symbol added.
Here's what we come up with:
Ab Db Dbm | Ab || Ab Abdim Ab Eb7 | ...
I IV iv | I || I Idim I V7 | ...
There is no diminished chord sense intended by Chopin in the first measure, even if it exists notation-wise it's an accidental byproduct, so there's no need to notate it. But in the 2nd beat of the 3rd measure we DO sense a diminished chord, so we notate accordingly.
CONCLUSION: there is NO general rule such as "to notate every chord every down/upbeat", but you should first attempt to pick up the significant notes and significant moments worthy of being labeled with a chord symbol.