What is it about the big D that doesn't work: reach holes, sealing those damned big holes (like the second one from the bottom), getting speed and ornamentation when everything is bigger or the amount of wind needed to play them? Each of these is a totally separate problem but all get worse as the whistle gets bigger.
If you don't have short fingers, you presumably dont have reach problems (although you may need to learn the "piper's hold" for best results. If you fingers are narrow, you may have troule sealing the big holes, but the piper's hold may also help (somewhat) as you use the thicker parts of the fingers on the big second hole.
Fluency of playing should come woth increased familiarity with the big whistles. Amount of air will vary a lot with brand and type. My first purchased big whistle, in the 1970s, was an Overton low F - nowhere as big as a D but it got me going and when I did get a low D the transition was not so severe. The early Overton were very "tight" - they had narrow wind channels and didn't use a lot of air - but needed good pressure. I have ever since preferred this to more open wind channels that can produce very mellow sound but need a lot of wind. This varies greatly between brands and, without knowing what type of low D you have, I can't comment.
BTW:I have, since the earlier incarnation of this thread, found that there are good acoustic reasons for the very wide spacing of low D whistles like the Overton. If I move the lowest hole up, or even tighten the grouping of the bottom three holes, I end up with an instrument that is not anywhere near as loud as one with the difficult fingering. At least I can position holes to suit the curve of the hand and, maybe fudging the size and position by minimal amounts, end up with a whistle that is a lot easier to play.