What works analogous to works of mercy do Buddhist monks practice? In Christianity monks and nuns practice works of mercy which include caring for the terminally ill or doing what St. Teresa of Calcutta did. What works analogous to what St. Teresa of Calcutta did 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.