Namaste Vijay ji,
In our tradition, we use Narayana dasa in all the divisional charts. Narayana dasa is a generalization of chara dasa. We take the karyesha in rasi chart and take his place in the divisional chart to start the dasa (e.g. position in dasamsa of the 10th lord of rasi chart is used to find Narayana dasa of dasamsa).
I have seen some people computing planetary longitudes in divisional charts and computing different Vimsottari dasas in different divisions, using the divisional longitudes. There is some basis for this. Parasara defined the exaltation and debilitation of some planets to occur in parts of signs, i.e. between certain degrees, and not in the entire sign. For example, if Mercury is in Virgo, he may or may not be exalted. He is exalted only in the first 15 degrees. But Parasara referred to planets being exalted or debilitated in divisional charts also! So case can be made that planets have longitudes in divisional charts also.
After all, computation of some charts such as navamsa-navamsa (D-81), navamsa-dwadasamsa (D-108), dwadasamsa-dwadasamsa (D-144) is based on treating the sign occupied by a planet in a divisional chart (e.g. navamsa or dwadasamsa) as if it was a sign in rasi chart and finding further divisions in it! Thus, there is basis for treating a planet's placement in a divisional chart as if it were a placement in rasi chart and finding a longitude for it.
If there is a divisional longitude, there can be a nakshtra too. Naturally then, there can be a Vimsottari dasa too.
However, we have to understand what the divisional nakshatra means. The rasi chart is based on physical zodiac and it shows the physical existence. The divisional charts are not based on the physical zodiac and are abstract. Similarly, the divisional charts do not show the physical existence. They show various abstract environments which influence one's physical existence (rasi chart). The rasi position of Moon shows the physical mind. Vimsottari dasa traces its progression with time. If you are not talking about the physical mind but talking about an abstract aspect of the mind (e.g. mind as applied to professional matters), will the same progression work? What is its meaning?
I did dabble a bit with divisional Vimsottari dasas, but I am happy using the normal rasi based Vimsottari dasa and judging the same dasa using various divisional charts.
But for those who want to do research, I have given the ability to find Vimsottari dasa using the divisional longitude in any divisional charts in the commercial version of my software.
BTW, I found the divisional nakshatras useful when analyzing transits using SarvatoBhadra Chakra. SBC is a fine technique for judging transits. Using the natal divisional nakshatras of planets and finding vedhas on them by transit rasi nakshatras of planets, I found very interesting results.
For example, my Jupiter is at 9Li46. He is in the 10th house in D-24, with a subhargala on arudha lagna. In D-24, this translates to 24Pi24 in D-24 and means Revati star. When Jupiter was transiting in Revati and Moon and Mercury had vedha on Revati from Mrigasira in SBC (the two planets associated with the 5th from arudha lagna in D-24), I stood state first in my state in intermediate exams and was selected for IIT-Chennai.
In my natal dasamsa, Moon is the only planet in the 11th from arudha lagna. He shows the earnings. He is at 28Aq33 and this translates to 15Sc30 in D-10. This means Anuradha star. When Mercury and Venus were transiting in Anuradha star in Dec 1993, I started in my first job. When Sun and Mars were in Asresha having vedha in SBC on Anuradha on 2002 Aug 12, I lost my job. The startup I was working for went into hybernation mode and laid off almost the whole engineering team. Please note that Sun and Mars are the 12th and 8th lords in natal dasamsa. Later that year, when Jupiter moved to Asresha and Venus into Visakha and both had vedha on Anuradha in SBC, I joined another job.
May Jupiter's light shine on us,