How to place the vertices of one mesh on the height of another mesh?

The solution was found independently.

 

There's a lot of ways to tackle this problem.

Probably the easiest to do this would be to make Mesh 1 a mesh collider and raycast against it from each vertex position to sample the height.

If Mesh 1 is a terrain it's even easier, Unity has a Terrain.SampleHeight method already.

You could also bake a heightmap from Mesh 1 and apply it to Mesh 2 using a shader or compute shader.

You could also do the mesh "raycasting" yourself and calculate the barycentric coordinates for each vertex in Mesh 2 on its complementary point in Mesh 1, then use those results to calculate the lerped height.

 

If it's a terrain, simply use SampleHeight.

 

Ah, ok. I'm assuming this is runtime procedural generation?

I'd still be tempted to use a compute shader, especially since it only has to be run once.

If not, however, I'd do what I suggested in my last point: calculate the barycentric coordinates for each vertex on Mesh 2 for its neighboring vertices on Mesh 1.