Let us look at the reconstructions (specifically, at Baxter-Sagart):
- 吏 (lì) 'officer' is [r]əʔ-s;
- 史 (shǐ) 'scribe' is s-rəʔ;
- 使 (shǐ) 'to send, to cause' is s-rəʔ;
- 使 (shì) 'to be sent as an envoy; envoy' is s-rəʔ-s.
I will append also the probably basic etymon, even though it is written with an unrelated character:
- 理 (lǐ) 'envoy' is m(ə).rəʔ (or probably just rəʔ for this word).
The relation between all the words is clear:
we have a basic root rəʔ 'to serve' (which, per Schuessler, is the same as 理 'to regulate, to administer'; Baxter-Sagart p. 144),
from which we get 吏 'officer' rəʔ-s with the ordinary
nominalizing suffix -s,
with the valency-increasing prefix s- we move to 使 'to send, to
and finally nominalize the latter as 使 'envoy' s-rəʔ-s ('one who
(The verbal meaning for 使 'to be sent as an envoy' s-rəʔ-s seems to be a later development when the Old Chinese morphology alrealy broke up at it became possible to just fluidly change parts of speech, per Schuessler p. 350; same for 史 'scribe' s-rəʔ, which looks like a direct change of parts of speech from 使 'to send, to cause, to employ' s-rəʔ).
From these on, the current pronunciations are all directly derivable. Thus, 吏 actually maintains the original consonant the best of all; all the rest in the group use the valency-inclreasing s- prefix and thus obtain the initial s-r- cluster, which is quite expectedly simplified to a retroflex version of [s], that is, [ʂ].