It seems that one is able to add a passing diminished 7 chord in between most diatonic chords going upwards, but going down, it doesn't always work.
For example, G > D > D#dim7 > Em sounds great, but G > Em > D#dim7 > D doesn't really work.
The list is longer, but in general, the passing chords tend to work going upwards only.
Dim chords can be thought of as dominant chords. D#dim7 works as a substitute for a B7 chord (as well as D7, F7 and Ab7). Try it - substitute any dominant seventh with a dim7 that's based on the dominant's third (or seventh). To find a suitable dim7 for leading to a D major, you want to substitute D's dominant i.e. A7. The third of A7 is C#, so --> C#dim7. Try G > Em > C#dim7 > D for your second example. Or e.g. G > Em > Edim7 > D. Or you can put the bass there too G > Em > C#dim7/A > D.
For a circle of fifths, try: Edim7/C > Ebdim7/F > Ddim7/Bb > Dbdim7/Eb > Cdim7/Ab > Bdim7/Db > ... etc.
To answer your question, why D#dim7 doesn't work in leading to D, it's because none of the four dominants that can be substituted with a D#dim7 is A7 that would lead to D in a V->I fashion. The four dominants that can be substituted with a D#dim7 are B7 (leading to E), D7 (leading to G), F7 (leading to Bb) and Ab7 (leading to Db).
Try these:
Or their minor variations:
Someone downvoted this, maybe they don't like to think of dim chords as dominants. :) But this is a very powerful way of handling dim chords, because it makes their identification and function easy to see, as long as you know how the most basic V-I i.e. dominant-tonic motion works. On the guitar you only have to locate the third (or seventh) of a dominant seventh (or secondary dominant or whatever you're doing) and play_any_ dim7 that includes that note (because dim7s are symmetric). This is a nice trick in soloing. Whenever there's a V-I motion: find the dominant's third and place a dim7 arpeggio pattern that has that note. For example in a G7 - C motion, outline a Bdim7 chord on top of the G7.
The same principle can be applied in this situation: How to choose between Cdim and C#dim? (Good and useful answers are downvoted, and the most upvoted answer doesn't really even answer the question. Someone here has a problem with dim chords?)
Why don't diminished 7 passing chords work going downhill? ...D D#dim7 Em sounds great but Em D#dim7 D doesn't really work
You can use a passing diminished seventh chord between descending roots, but reversing the ascending root version won't work. Generally, reversing chord progressions doesn't work. Or, perhaps more correctly stated, reversing progressions changes chord functions.
What you need to do is use a passing diminished seventh chord build on the leading tone of the target chord. Use a D# diminished seventh to go to an E chord. Use a C# diminished seventh to go to a D chord.
You can then use chord inversions to create smooth voice leading. As a general rule, if you have the leading tone in the bass put the 3rd of the diminished seventh chord in the top voice and vice versa, and then move bass and treble in contrary motion.
I disagree with the premise of this question. In many songs, diminished chords do pass downhill in that manner exactly; however, it's more often done in jazz-influenced styles of music than classical. As an example of nearly that exact chord progression, take "Sweet Sweet Canyon" from the Mariokart 8 soundtrack. In C major, (simplified)
C E♭°7 Dm7 G7
C E♭°7 Dm7 G7
C Em7 F G7
Em E♭°7 Dm7 G7
At the end there is G Em E♭° Dm, proving that the upper diminished passing chord does in fact work. The Dm could just as easily be D, and there are thousands of other examples of this diminished passing chord resolving down, if not tens of thousands.
Existing answers are pretty much valid, but I'd add that the upper leading tone can resolve down in a diminished passing chord movement. It's probably an acquired taste, as is much of jazz harmony, but it certainly works well enough. The other answers also mention that the lower leading-tone diminished chord can be viewed as part of a secondary dominant. While the upper one can't be sensibly explained as a secondary dominant, it still functions due to the half-step and thus doesn't need to be the vii° of anything.
We've had a recent discussion about upper leading tones and lower leading tones in diminished passing chords before on this site, and I hope this'll help you reckon with the strange world of diminished chords.
More "upper leading tone" sources: A definition, Fm-C, something closer to home, analysis, and this essentially says what I'm saying without actually using the term
I can think of plenty of examples where descending diminished passing chords do work, in Brazlian music they're pretty common.
Most notable one that springs to mind is in "Canto do Xango"
| Em | Em/D♯ | Em/D | Em/C♯ | | Em/C | B7 | A♯° | Am | | G6 | B7/F♯ | Em |
also seen as:
| Em | B7/F♯ | G | Am | | C | B7 | A♯° | Am | | G | B7/F♯ | Em |
There's also "tempo de amor" from the same album which has:
C- B° E♭6/B♭ Aø A♭7 G+7
(played here because it's actually pretty hard to make out the changes on the original record so I played them slow and clear https://instaud.io/3ECq)
Corcovado has some funny extentions but uses descending diminished movement too:
simplified it's
Am6 > G#dim > Gm7 > C7 > FΔ
(recorded here with the voicings above to prove I'm not bullshitting)
https://instaud.io/3ECc
Using your example, D♯o has notes shared with B7, namely D♯ F♯ and A. This makes the change to Em sound natural - V>I. Or, if you like, including the 'C' note, could be called B7♭9
Going the opposite way, towards D major, that same D♯o has only an A note which would point towards the D chord.
