Exactly! That is the type of thing I was thinking, done with whatever fencing material you fi d appropriate. The neighbors may indeed see it, and may even object (aka whine) but if the dogs aren't on your property they aren't at risk of being caught, right?That I can do making it like a large fishing weir like shown below except made of goat panels. Opening would face where dogs come through fence. What might make is really impactful is having it setup so dogs enter as soon as they cross property boundary and neighbors can see it and recognize what it is for!
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As far as containment/transport. Do you have the ability to build a hog crate type box? With a sturdy box wall you can use a simple ramp and come along to drag the box up into a truck or any vehicle with a rear cargo area (even my nissan rogue would fit a large crate). Make the door a straight slide up type hatch so the crate can be placed directly against another cage, the door slid up and the dog shifted to the new cage.
You can probably build a substantial short term crate for not a lot of money, just the "head scratching time" as I call it and some sweat equity.