The form of the ablative case suffix in Mongolic is a well isogloss separating all modern Mongolic varieties and all Middle Mongol sources save Muqaddimat al–Adab. Middle Mongol reflects *-(a-)ča, while modern languages (and MA) show *-(a-)sa. Thus, irregular deaffrication of *č in this suffix is taken to be a Common Mongolic innovation.
There are no external comparisons that could hint to a further etymology of the ablative suffix. Khitan has an unrelated ablative in <e(n)d.ii>, homographical with dative+genitive. Sometimes a comparison with the Turkic equative (likely an original prolative) *-ča is made, but the semantics are off. Poppe also claims cognacy to Manchu či, but Jurchen data (Kiyose 309) shows that the original form was ti and the similarity probably fortuitous.
The alternation between *s and *č doesn’t seem to otherwise occur except through assimilation of *s to the following *č (also found in Turkic), neither in roots nor in morphology. It does however remind of an alternation between *-gs- and *-s- found in some suffixes (past participle, adverbial directive, …), although *-s- variants haven’t reached fixation in all modern Mongolic languages.
As I am generally suspicious of late innovations spanning from Mogholi to Dagur, I wonder if it could be that the *-ča form itself is somehow innovative. Ablatives are likely to be built or extended with locative suffixes, this happened multiple times with this very suffix, thus we have forms such as -a-ča and -da-ča. I would thus posit that maybe *-ča itself is built as *-d-sa, with modern languages preserving bare *-sa.
The first element is the morpheme found in dative-locative forms -dur/-du/*-da, where *-r and *-a are known dative-locative suffixes. The *-u- of the first form could be explained as epenthetic, but in the second form it is harder to explain in the light of the third form, Mongolic has no synchronic rule to the effect of -u + a- = -a- (although perhaps that should be investigated).
A problem with this is that Mongolic follows strict “one onset – one coda” restriction, so *-dsa should have an epenthetic *-u- after consonant stems, like other cluster initial suffixes, which it doesn’t. One explaination could be that this pattern was analogically removed after the coalescence of *-ds- into *-č-, but perhaps this distinction was still marginally felt in some dialects of Middle Mongol which led to popularization of the extension in *-a-ča.
We could also consider the possibility that precursors of modern varieties also had *-(a-)dsa- which was phonemically different from an affricate, and dropped the *-d- as *-g- was dropped in *-gsan and *-gsï. A similar phenomenon is found in some dialects of Serbo-Croatian that occasionally drop /d/ in /ds/ clusters, despite maintaining a phomenic /ʦ/.
 As Poppe notes, Naringhol Mongghul has -za /tsa/, but this is more likely to be a secondary innovation as sporadic affrication of *s into z occurs also in other suffixes and words. A similar thing seems to be going on in Dagur (and Solon), but not in this morpheme.
 Dagur has deaffrication of *č into š which seems (pace Nugteren) regular, initial *č is deaffricated before an *I, with exceptions due to pre-breaking.
 Janhunen claims also the existence of bare -d in adverbial “uri-d” ‘before’, but this word is undoubtedly better reconstructed as *urïda.