Dim 7 movement

Hi Lee. Hope all is well. Ok, My question is when practicing Dim7 chords when I move them up to a new root  ie- Cm to Em , would the next movement be to Gbm then to Am?  If so ,Then I understand what minor 3rds mean. But although the shape stays the same does the new root note need to be on the same string as the first chords root note? I know the correct chord formula (1,3b,5b,7bb) so my question is purely about the movement and the root notes position. I’m afraid to try and figure this out alone and possibly waste time with incorrect thinking. Thanks in advance, Sean