The Latin quip was derived from Aristotle's History of Animals, where it is rather specific to the context of the "scale of life". Hence Linnaeus's and Darwin's references are more on point than more sweeping post-Aristotelian generalizations about "most physical processes", some quite remarkable in how obviously false they are. It should also be said that Darwin's use had a very specific target, Georges Cuvier and his paleontological catastrophism (which was already replaced by Lyell's gradualism, Darwin's inspiration) extended to biology. Here is Aristotle:
"Nature proceeds little by little from things lifeless to animal life in such a way
that it is impossible to determine the exact line of demarcation, nor on which
side thereof an intermediate form should lie. Thus, next after lifeless things in
the upward scale comes the plant, and of plants one will differ from another
as to its amount of apparent vitality; and, in a word, the whole genus of plants,
whilst it is devoid of life as compared with an animal, is endowed with life as
compared with other corporeal entities. Indeed, as we just remarked, there is
observed in plants a continuous scale of ascent towards the animal."
Already Pliny the Elder in Naturalis Historia extended Aristotle's scale to scala naturae, and this is the "ladder" Kant refers to. Plotinus and Christian authors took it all the way to the Heaven as a "great chain of being" much earlier, see Archibald, Aristotle’s Ladder, Darwin’s Tree.
As for the quip, "Theutobocus" is not the author name, the reference is to Discours véritable de la vie, mort et des os du géant Theutobocus by a rather obscure Jacques Tissot (Lyon, 1613). The "bones of giant Theutobocus" apparently referred to bones of a mastodon presented as those of a human giant in a hoax. That a French work inserts a Latin quip, calling it an "axiom" no less, suggests that it was already in circulation by then. Classical and Foreign Quotations, p. 209 confirms it:
"Tissot is quoting an old and well-established principle of physics. "Operatur natura", he says, "quantum et quandim potest, sans neant moins faire aucun sault ab extremis ad extrema. Natura enim in operationibus suis non facit saltum". His contemporary, Sir E. Coke, applies it to law:"Natura non facit saltus, ita nec lex". Coke upon Littleton, pp 238b, 239."
And then it goes to Leibniz and Linnaeus. Wicksteed in Studies in Theology, p. 30 adds the following:
"I have not been able to trace it further back in the form of an aphorism, but the principle is taken for granted by Dante as holding good in the physical universe, and is applied by him to the intellectual order."
A lengthy quote from Dante's Convivio, III.7 follows, where the "great chain of being" is on full display:
"And since in the intellectual order of the universe the ascent and descent are almost by continuous gradations from the lowest form to the highest and from the highest to the lowest, as we see in the order of beings capable of sensation; and since between the angelic nature, which is intellectual being, and the human nature there is no gradation but rather the one is, as it were, continuous with the other by the order of gradation; and since between the human soul and the most perfect soul of the brute animals there is also no intermediary gradation, so it is that we see many men so vile and in such a state of baseness that they seem to be almost nothing but beasts. Consequently it must be stated and firmly believed that there are some so noble and so lofty in nature that they are almost nothing but angels, for otherwise the human species would not be continuous in both directions, which is impossible."