For instance that AI Civ that is heading to the Barbarian Camp beyond your borders that you can’t see yet.
Good! I don't want to have to have a unit out fighting barbarians and losing.
Also, for that matter, why did I have barbarians at roaming or higher in the game to begin with? I guess they provide a little extra gold if defeated, but there's not enough benefit. The AIs can get more benefit from barbarian camps with enough free units and a free settler or two also, I think. Barbarians make things less predictable. Also, I thought about playing for a histopgrahic game for a COTM somewhat recently. But, the map had barbarians. I realized I would need some units watching territory until the end of the game. Maybe I've missed something. But, why do you want barbarians? Do the benefits outweigh the hassle of them?
hey start trickling in a line of multiple units. You kick them out, but they start marching right through again!
I remember reading an article for winning on Deity a longtime ago by Drakan which suggested just letting them walk over your land. At the default difficulty levels where the AIs have a free settler, your empire is wimpy early. Why aggravate someone who can bully your empire? Even on 'least aggressive' you can have Annoyed AIs by the end of the ancient age. Everytime you tell an AI to get out your territory, that decreases their attitude towards you. From what I can tell AI attitude affects the cost of workers and the exchange rate on some difficulty levels (though it's fixed on least aggressive). Good AI attitude also helps to decrease the probability of sneak attacks, and can help to win a diplomatic victory. So, one very much need a strongly compelling reason to tell AIs to get out of one's land, if one thinks things through in my opinion.
If it's war for trespassing, that decreases the difficulty range for a game. No matter the tactics, I don't want to imagine a "trespassing means war" constraint on a Sid pangea game. I'd guess that in 80%-90%, if not 98% of cases, that would be a loss. So, I can't really recommend such a constraint as "trespassing means war", unless it's deliberately a constraint to make the game more difficult.
an ROP to get them moving of the way fast
I don't think so. There exist some cases where I've outsettled AIs that walked into my territory. Then the AI will turn around and go the other direction. Though, you might want to consider RoPs for other purposes... though I've often tried to avoid RoPs early to slow down AI settling on high levels come to think of it.
just let them slowly pass through blocking multiple tiles of your territory
Yes, that can work! I would say that it's not always necessary or optimal. Like I believe that if you can produce a worker in 5 turns with minimal to no shield overrun, a worker is better than preventing AIs from walking through your lands. But, if it seems like a decent time to put out a unit, blocking them off might work. Also, once they get Map Making blocking with land units won't work as much. So, of course, it's not something to get relied on.
Additionally, AI trespassing with settlers sometimes can get leveraged beyond just getting some slaves. Like let's say you play on a level where the AI has a good amount of free units and a good growth rate. Early on, the military advisor will rate them better than your empire, possibly until the end of the ancient age. Now, let's say Elizabeth from the other side of the continent sends a settler your way. And you're behind in tech.
Well, then set your tax slider to 100% temporarily. Maybe also change all citzens to tax collectors. Then ring up Elizabeth. Tell her you will pay gpt only for whatever technology she has, or the gold she has in her treasury (if it's just gold, you probably don't need to lift citizens for tax collectors). Now since she has 2 units in your territory, tell her to get her units out or declare war. Well, she will declare war (or at least will if she's Furious also). Then you get to keep the gold or technology. And she's on the other side of the continent, and not really a threat (though maybe she can sign military alliances).
But note the above isn't necessarily a thing to do as soon as an English settler-spearmen pair walks into your territory. It often enough can work out as more beneficial to let their units walk around for a while.
So, I'd say that the trespassing doctrine is more like "get good stuff for free from trespassing AIs, when you can handle a war against that AI, and you don't mind putting out units." Kind of like what Moonsinger suggested she did in GOTM games years ago. In such cases, I certainly don't find trespassing annoying, but instead a good opportunity.