I've seen the What Programmatically Triggers Magento 2's Shipping Rates to Update question, and also read related @AlanStorm's post on his website, but I still don't understand, what I need to add into my custom JS code so that the shipping rates block is updated.
i.e. it automatically updates when postcode field is changed. But even if I manually fire jQuery
$(postcode_field).change() trigger, it works only if postcode really changed.