Reaction of carboxylic acid with phosphorus trichloride
A plausible mechamism of this reaction is discussed in a paper by Xiao et. al1 where they tried to react benzoic acid with $\ce{PCl3}$ to get acyl chloride. They noted the following mechanism:


Although the detailed mechanism was not clear, a possible mechanism is
proposed [...] the first step should be a reaction of the acid with
$\ce{PCl3}$, probably via a four-center transition state A, with
carbonyl oxygen attacks the electron-deficient $\ce{P(Cl)}$ while $\ce{Cl}$
attacking the electronically deficient carbonyl carbon. Formation of a
$\ce{Cl-C}$ bond and $\ce{P-O}$ bond (A’), further $\ce{C-O}$ bond cleavage and
hydrogen transfer took place to give the products B and C.
Product C then reacts further with the acid to give the
corresponding chlorides and $\ce{HP(O)(OH)2}$

This product C will again react with carboxylic acid two times so as to replace all the chlorine atoms and ultimately forming phosphorous acid (actually phosphorous acid technically exist as $\ce{HPO(OH)2}$ called phosphonic acid) So, the final reaction would be three moles of carboxylic acid reacting with $\ce{PCl3}$ to form three moles of acyl chloride and phosphonic acid thus fully utilizing the three chlorine atoms in $\ce{PCl3}$. The major advantage of this reaction is that there is only one side product i.e phosphonic acid which is non-toxic and easily removed by filtration whereas reaction with $\ce{PCl5/SOCl2}$ gives two side products making the reaction more volatile.
Reference

Atom-efficient chlorination of benzoic acids with PCl3 generating acyl chlorides, Jing Xiao and Li-Biao Han, Journal of Chemical Research, 2019, Vol. 43(5-6) 205–210 (PDF)