S. initial [x-] became [h-]; [x-|x{lr}-] > [h-|{l̥r̥}-]

At all stages of the languages (Sindarin, Noldorin, Gnomish) an initial [x] (χ) developed into [h], as noted by David Salo (GS/§4.130). Tolkien mentioned this change in his notes on the Noldorin use of the Feanorian Alphabet from the 1930s:

S. medial [mf], [nθ], [ŋx], [lθ] became [mm], [nn], [ŋg], [ll]; [-{mf|nθ|ŋx|lθ}-] > [-{mm|nn|ŋg|l̥l̥}-]

S. [ð] vanished before nasals; [ð{mnŋ}] > [ø{mnŋ}]

For both Sindarin and Noldorin, in compounds where a voiced dental spirant [ð] came in contact with a nasal after various vowel losses, the [ð] vanished, as noted by David Salo (GS/§4.119). There are several examples in both Sindarin and Noldorin:

S. nasals vanished before spirantal clusters; [{mnŋ}{fθxs}{lr}] > [ø{fθxs}{lr}]

S. final [mɸ], [nθ], [ŋx], [lθ] became [mp], [nt], [ŋk], [lt]; [-{mɸ|nθ|ŋx|lθ}] > [-{mp|nt|ŋk|lt}]

In Sindarin and Noldorin, where combinations of nasals and voiceless spirants (mph, nth, nch) became final again after short final vowels vanished, the spirant became a stop again (mp, nt, nc). The same is true of the combination lth (becoming lt) but not rth. Tolkien described this change in a couple places:

S. [ls], [rs] became [lθ], [ss]; [ls|rs] > [lθ|ss]

In Noldorin it seems primitive [ls] became [lθ], but [rs] became [ss], as noted by David Salo (GS/§4.83, §4.183):

S. [nr] became [ðr]; [nr] > [ðr]

In both Sindarin and Noldorin, where n and r came into contact in compounds after the loss of vowels at morpheme boundaries, the n became dh. Tolkien actually mentioned this rule in The Lord of the Rings appendices:

S. [œi] became [ui] or [y]; [œi] > [ui|y]

One open question is how the diphthong [œi] developed in Sindarin, assuming it occurred at all. This diphthong is relatively common in the Noldorin of the 1930s (where it became [ei]), arising in the phonetic history of plural forms of words with a base vowel o, where the raising of [o] to [u] was usually inhibited in the final syllable of polysyllables: