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Does that mean I can march thru all "his" vassal provinces that are not at war with me & not cause them to declare war on me?

Except in the probability of him calling up HIS vassals troops, at which point they'll be under his command and you'll have to fight through them to get to your destination (the ruler's demesne).
 
In that case would I be at war with them?

Or could I just not stop & declare war on them myself & take those provinces?

You wouldn't be at war with them personally but also you aren't fighting them personally. You're fighting their liege who just happened to control those troops.

Just like any of DOW, you'd need a claim on a title to take their provinces, unless they are religious enemies, then you wouldn't need a claim and can take their provinces.

When I'm fighting a huge Muslim/Pagan power, that's how I usually go about it. I'll DOW one of his vassals (which brings the Liege into the fight), he calls up all of his vassals and starts moving them towards me. Then I do a sweep around his front (leaving defenses) and start picking off his vassals. Thereby weakening his main armies because their troops disband as I conquer each one, cutting his power out from beneath him. This all depends on how strong personally he is. If he's a one province Duke (and I'm a King), then I'll make a straight line for his province, vassalize him and bringing his entire Realm into my Realm in one go. If he's a King with load of personal demesne provinces, then it's taking all of his vassals away, then getting a Peace out of him so I can take his personal provinces over a series of wars.