Wiktionary allegates that, for the Latin mittō (“to send”),
The semantic shift from “send” to “put” probably occurred in Vulgar Latin.
What semantic notions underlie “send” and “put”? I can’t brainstorm any relationship between the two, even after reading this word map or narrative.
The word in the title, mittō, was the one used in ancient times to name sending, others would also have been valid to name something very different then from what it is now. 
Shame on me, I cannot account for the semantic transition (quite a jump, actually) in the meaning of mitto from “send” to “put” (cf. Italian mettere (put), having no immediate connection to “send”).
But apparently such a transition occurred well before the onset of Vulgar Latin: on the prestigious Campanini-Carboni - Latin-Italian-Latin dictionary, the entry mitto takes a whole page where many bordering (and less bordering) meanings are quoted, ranging from “send”, to “release”, to “produce”… to “put”. Under the latter it cites quotes from Seneca (4BC-65AD), Ovid (43BC-18BC) and Virgil (70BC-19BC).
I agree that the above casts more darkness than light on your question and suggests that the semantic shift you’re intrigued by should be looked for much further back in the past - perhaps even before the dawning of Latin as a language.
Perhaps you’re too intelligent to find an answer. I’m not. 
(Fools rush in, …)
A “put” could be described as nothing but an extremely short “send”, a direct one that lacks any intermediaries. If I “send” you a letter across the world, or if I “put” it on your desk with my hand, my goal is the same in each case.
Hmmm… I tried to send my glass of wine on the table but the delivery took four days and it crashed on arrival… 
If a language requires its verbs to be foolproof, the administrators of that language receive an award: a note of congratulation “from your friendly local EU, for making our less-popular policies look reasonable by comparison”, which (“that we may honour your assistance in propagating our shared aim of building bureaucracy for its artistic value”) is accompanied by the obvious symbol for “propagating the EU”, the Golden Brussels Sprout. The sprout is “golden” because, like the best bureaucracy, it has been grown in the dark. And yes, they have a DOP for those.
Allegedly in PPPL (Preprotopaleolithic) there were six verbs altogether: “slaigh” (kill), “mkh” (make), “mawf” (move), “gnahm” (eat), “zzzh” (sleep) and “phkk” (to denote other activities or to express disappointment).
In the ensuing millennia though, as the gestures used to convey the fine nuances were gradually abandoned, those basic verbs sprouted into a number of specialized variants, thus greatly enhancing the communication skills of homo sapiens.
Reversion to the mean?
Reversion to the meaningless? 