Laurence Payne's answer is pretty much the gist. But just to address a few specific questions:
Does he mean that every chord note should proceed to a harmonically
correct note in the corresponding metrical position?
Not necessarily. In very regular broken chord patterns (e.g., Mozart-style Alberti bass), it often happens that most notes do create lines within the same metrical positions for each beat. But that's only true with a very regular pattern. In most cases, those patterns will need to be broken sometimes to create a better pattern overall, even for Alberti bass.
For example, let's imagine you are in C major. You have a broken G7 chord with some pattern. When that chord resolves in the next beat with another broken C chord, there will be a C generally in the same octave that the B was in the G7 chord. And there will be an E in the same octave that the F in the G7 chord was. Often these will occur in the same relative metric positions, but that's not necessarily the case. You just want to hear a resolution of those two tendency tones somewhere in the following chord, overall creating a sense of B-C and F-E voice-leading.
But how would that work if the chord were broken in notes of unequal
Doesn't matter. You adapt as necessary.
I'm also confused as to what should happen to chords that are broken
in a 'mixed' manner (with both ascending and descending motion)
or chords with repeated tones.
Do you mean "repeated" as in the same note repeats multiple times within a pattern (e.g., C-G-E-G in Alberti bass as "one chord") or do you mean "repeated" as in the note appears in multiple octaves within the broken chord?
Either way, it doesn't matter. As Laurence Payne said, just imagine that each broken chord is written as a single vertical chord. Then apply standard voice-leading rules (like resolving leading tones, no parallel fifths, etc.). Then break the chords up again.
In some cases, metric position does matter more. For example, parallel fifths will be heard in broken chord patterns more clearly if they occur on relatively strong beats within the pattern. On offbeats, they may not be noticeable. The metric placement is mostly relevant for those sorts of issues. But for other issues of voice-leading (e.g., leading-tone resolution, continuity of line), metric placement can be more flexible.
If you still don't get it, I'd recommend taking a look at Bach's solo suites for cello or violin to see this sort of thing in action. Bach is much more flexible about metric placement for the voice-leading in broken patterns than, say, Mozart's Alberti basses. But it's good to compare various broken chord styles in different eras to see how different composers handle this.