I came across すすんで行く and さいて行く in a vocabulary list, and it was immediately obvious that there was probably some sort of grammatical construct involving て行く that was in use here. I don't really know what this grammatical construct means though, and it's apparently something different than just compounding two arbitrary verbs together with base te.
A quick Google search brought this lesson up, and he starts to explain what the construct means (as well as て来る), but he doesn't do a very good job of it. He also starts to deviate from the special grammatical construct towards just the base te verb compounding, and many of the other search results are focusing almost exclusively on the compounding.
I vaguely get the notion of this construct meaning to get into the state of doing something (て行く) or to have come into the state of doing something (て来る), but I really don't understand it.
Could somebody please explain this? I'm mainly wondering about the general grammar being used here, but I also kind of wonder about how it applies to these two words, since they're apparently decent examples. Thank you.