Edge and Vertices 'Not Connected'

Hello everyone, I’ve managed to create some situations on one of my models where the edge and verticie do not appear to realize they are connected to each other. In the picture, I’m dragging a face up and to the side but notice that some of the vertices are not ‘coming’ with it, they just sit in where they were.

So I’m looking for how to fix that problem. As far as I can tell, the vertex is on the edge and I’ve tried ‘snapping’ them together, but it didn’t make any difference.

Any ideas are appreciated, thank you for reading.

Here’s one method to fix that.

I see, thank you very much for the video.