Oxidative chlorination mechanism (sulfide to sulfonyl chloride)
As Ben Norris already pointed out, these processes are typical a bit complex. It is however conceivable that the direct oxidation of the thioether 1 is among the first steps involved. 



While the oxidative cleavage of PMB (= 4-methoxybenzyl) ethers, often applied as protecting groups, is assumed to take place in the methoxy-substituted arene, here the electron transfer mostly likely proceeds from the sulfur atom, leading to the sulfur-centred radical cation 2.

In the presence of a nucleophile, this species is likely to break down en route to the radical 3.



Whether a disulfide, formed upon dimerization of 3, will take part in the further course of the reaction is beyond these musings.