I would suggest keeping this vine off your trees. OK for trellis, garden walls and things but not on trees.
Just because it's native doesn't mean it's going to get along great with another species. Think wolf v/ rabbit. Or better yet think about grapevines which are native to parts of the country and can be destructive to trees. They'll complete for sunlight in the canopy, nutrients in the soil, cause branches to be deformed or die altogether. Like I said... garden walls, chainlink fence and trellis, lovely for butterflies. But I'd keep it off the trees.

Let me begin this argument (it will be!) by saying that some vines only attach to the exterior bark of trees and do not become a problem until they overtake the tree's foliage and compete with it. Other vines actually grow roots through the bark to the cambium and become parisitic then begin to ursurp food storage to the root system of the trees. Which vines do what? Here we go.
hortster

I'm not going to argue that crossvine is or isn't harmful as I'm not all that familiar with it but I'd still keep it off the trees and use it somewhere like a trellis, wall, fence.

It is IMPERATIVE to be able to see the structural integrity of yard trees.

In the last 7 years I've removed two large white ash trees from my property. Both because of splits at main branches. IF either was covered by ivy it would have been impossible to see this and poof, by one one would have fallen on my house (or Mustang! or wife).

Now if its in a naturalized area let nature take its course to some limits.

I suspect though natural control measures which limit the spread of vines in woodlands are not present in your yard. Fewer vine eating animals, less fire for a couple. So you'll need to play the control part or else the vine stands the potential to take over.

How do you keep crossvine from spreading? What kills it?