Is it possible to move snapped verticies together? This feature is available on QGIS, I believe it is the magnet tool.Snapped Polygon
Vertices unsnap when moving
Yes. With a Map Topology enabled in ArcGIS Pro, you can move coincident nodes quite easily.
Not sure what you mean by moving 'snapped' vertices together.
You can hold the shift down and select (or drag to select) multiple vertices and they will all move together.
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