How to Move Vertices from a Mesh to a Specific Location

I have a dynamic mesh that is mostly comprised of rectangles. So far, the actual system has gone smoothly. I'm getting to a stage where I want to start polishing the actual meshes off, and the topology is part of that. As I have it now, the meshes aren't able to be placed exactly. That is, there is a gap between the meshes--a very minuscule one at that.

In the past, I've done similar things outside of Unity. Basically, what I'm looking to do specifically is move a vertex near another vertex so that they are at exactly the same position. At the moment, I've already implemented a spatial hashing algorithm that can sort the hashed vertices into buckets and then I can get similar vertices by iterating over the buckets. However, I'm stuck at the part of actually determining and moving the vertices into their exact locations.

I'm assuming that I need to somehow take the two vertices in the bucket and make them exactly the same objects. How would/should I go about this in particular? On a side note, is there a better option than spatial hashing for Unity and this particular situation?

Here's some code outlining my general pipeline for this algorithm as of now:

 // Loop over all the vertices in a mesh
 foreach (Vector3 vertex in meshVertices)
 {
     // If there is a matching vertex somewhere within a given
     // threshold in the table, returns true.
     if (SpatialHash.DoesTableContainSimilar(vertex, 0.001f))
     {
         // Retrive similar hash here, and then change the vertex
         // location to match that of the other
     }
 }

As of now, the problem remains of how to actually take that other hash and turn it back into a Vector3, and then moving both of the vertices to the same location. As it stands, the actual Vector3 object is stored in the key, value pair in the hash table. Hypothetically, could I just take that other Vector3 and Lerp them until they meet a certain point (i.e. interpolate them), and then set both values to the new, interpolated values?