In my experience*, checking in for a domestic flight is the same whether one is connecting to an international flight, connecting to another domestic flight, or flying non-stop to one's destination. They will check your passport at that point**, and any visas that your eventual destination requires, but that only adds seconds to the process so you can ignore it in your planning. The only thing that matters is if you are on time for that flight, and domestic rules apply there. If you're at the gate in time for your domestic departure with a valid boarding pass for that flight, nobody is going to look at your complete itinerary and say "we saw you walk in here ten minutes after the deadline for an international departure, so we won't let you on the plane."

What they do at your international gateway varies, but that has nothing to do with when you have to arrive at your initial (domestic) departure point. If anything, that might affect the time you need for your connection, but your airline took that into account when they established the MCT (Minimum Connecting Time) for domestic-international transfers at that airport.

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*Which is fairly extensive, and some of which is exactly a week ago today.

**You have no control over where the airline check your passport. They will check it when and where they want to, perhaps more than once. It is not a question of your convenience. You can't say "Please don't check it now, I prefer to have it checked later." You also can't say "Somebody else already checked it at some other airport." Neither statement will make the slightest bit of difference.