Gunpowder Empire

Jeremy Solters is a teenager growing up in the late twenty-first century. During the school year, his family lives in Southern California - but during the summer the whole family lives and works on the frontier of the Roman Empire. Not the Roman Empire that fell centuries ago, but a Roman Empire that never fell: a parallel timeline, one of an infinity of possible worlds.
For in our timeline, we now have the technology to move among these worlds. Some are uninhabitable; some are ghastly, such as the one where Germany won World War II. But many are full of resources that our world can use. So we send traders and businesspeople - but to keep the secret of Crosstime Traffic to ourselves, these traders are trained, in whole-family groups, to pass as natives.
It's a lot of work, especially since they're not willing to own slaves like everyone else in this version of Rome. And they spend a lot of time dealing with the local rules and regulations, where unofficial clout matters as much as official status and almost as much as money. Still, most of the time it's reasonably easy for the family to do good business, trading multigadget pocketknives and elaborate windup pocket watches for wheat.
Then Jeremy's mother gets sick - really sick, the kind you can't cure with antibiotics. Both parents duck out through the gateway for a quick visit to the doctor. But while they're gone, the gateway stops working. So do the communications links to their home timeline. Jeremy and his sister are on their own, the Lietuvans are invading, the city is besieged, and there's only so much you can do when cannonballs are crashing through your roof.