What works analogous to works of mercy do Buddhist monks practice?
I believe monks are enjoined to do the same for each other -- Kucchivikara-vatthu: The Monk with Dysentery
But  not to act as doctors for lay society in general -- reference

Thus, bhikkhus should not become doctors, full-time nurses or attendants to invalids. They were supposed to live a simple, unencumbered life sustained by sufficient nutriment, appropriate medicines and advanced health standards for the full-time pursuit of spiritual liberation.&quot;
&quot;When the Buddha referred to tending the sick, he was referring to fellow monastics. The Commentary,... [has that] a bhikkhu may prescribe and supply medicine to... his parents, to those caring for his parents, to lay-attendants of the monastery and to those residing in the monastery preparing to ordain; a bhikkhu may also [etc.]

Some will act like chaplains occasionally, if invited to -- visiting someone terminally ill and their family in hospital.
Other &quot;acts of mercy&quot; are a fairly rigorous discipline of being harmless, including not killing animals or insects -- and accepting charity (on the theory that it's good for laypeople to practice charity) -- and teaching the doctrine -- and so on, carrying on the Buddhist discipline and relaying the doctrine.

What the Missionaries of Charity did or do is a bit unusual among Christian monastics, though as the French proverb says, &quot;If they didn't exist one would have had to have invented them.&quot;
There may be some analous organization among Buddhist monastics, here for example -- Fighting addiction at a Thai monastery -- that's maybe a close parallel but unusual/atypical too though.