Modal Augmented Sixths
The augmented sixth chords have more generality across the modes since they can be derived from major or minor triads and in fact the six-five-three is an enharmonic variant of dominant seventh chords. The resolution of the augmented interval signals the difference with the seventh chord since the augmented note resolves upwards while the seventh resolves downward typically. The augmented sixth chords usually progress to V whether based on II or IV, but can also go to I or III.
Note that the Dorian progression involves the minor V and has one whole-step relation rather than all half-steps as in the Ionian augmented sixth progression to V while in the progression to Dorian III (major) there is now a repeated tone that in the Ionian mode moves by half-step. The Phrygian progressions are similar to the Dorian except that the progression to V involves a diminished triad. The Lydian mode already has the augmented fourth so the chromatic alterations would involve the flattened sixth scale degree (and seventh in the six-five-three chord). The voice leading is similar to the Ionian since Lydian V is major and III is minor. The reader should compare the sound of the regular Lydian IV - V - I progression with the six-five-three substituted for IV. The Mixolydian has a minor to minor six-four-three progression and a major to minor six-five-three progression to V, but Mixolydian III is diminished. The Aeolian has the minor to minor six-five-three to V progression but the six-four three chord is built on the diminished Aeolian II. The progression to III is similar to the Dorian. The Locrian progressions to V and III are similar to other modes but of course the tonic triad is diminished. The augmented sixth chords do provide a bit more momentum to the Locrian V - I progression.The advantage of the augmented sixth chord is that it can be produced via melodic considerations more variedly than the typical dominant seventh chord and it can therefore operate in modes not conducive to the formation of such secondary dominants (e.g. Phrygian). In the major-minor system, the uniformity of the harmonic organization led to the inclusion of remotely related chords to provide some variety, particularly in the cadences. In the modal system, melodic and harmonic variety stem more directly from the scales, but result in less tightly wound harmonic structures. Thus, these augmented chords or their voice leading equivalents can buttress a dominant or tonic without forcing the mode into the major-minor system.