get means a number of things, on its own and with a gerund. One of its meanings is to begin.
"get doing something" is colloquial speech in AmE, anyway and is the same as:
"get to doing something". He got to cleaning the steps and forgot his worries.
get to doing something is a higher register and not as colloquial, regional or dialectal.
He got cleaning the steps. means: He began to clean the steps.
get going actually means: start going or more idiomatic: start to go somewhere, leave a place
get working on it means: start working on it
get moving: start moving
He got moving on the project yesterday.
Generally, therefore get + (to) gerund generally means to begin or start something. That is what seems to govern it. So, if it fits with the idea of beginning or starting, it might very well work.