On a public fast you only do the liturgical changes (like Avinu Malkeinu) if you have at least six actual fasters present. If you don't have that quorum at the beginning and therefore start doing a regular weekday service, and then more people come in, do you change over? What about if you had already begun the torah service with the regular daily reading (assuming the fast day was also a torah-reading day in its own right, like today was)?
Does it matter if there is uncertainty about the new arrivals, so you don't know if you definitely have quorum now?
(This happened at the shacharit I attended today, though not during the torah service.)