I do that sort of thing regularly. That's why I prefer coated strings on my acoustics. They don't make as much noise when I slide.
changing between chords which don't use the same shape
the easiest thing to do technically speaking is to slide on whichever strings the two shapes have in common. As Dr Mayhem suggested, the root, third and or fifth (as applicable) will sound best.
There are sometimes cases where the two chords do not share any of the same fretted strings. In that case - what I do is form the new chord in one position on the fretboard and slide the chord shape up to the new chord. If you are sliding from an open position chord, you can play the open strings that will be fretted in the new chord shape and quickly "hammer on" the new chord shape in first position and quickly slide to the new position. This happens very quickly all in one motion (strum - hammer - slide).
For example, If I am playing a first position G chord using 320003 voicing and I want to slide to a D7 played as XX453X - I will strum the open D G and B strings and quickly hammer the XX453X shape on at first position (XX231X) and slide that shape up to the XX453X all in one motion. You end up with your target chord ringing out at the end - even with just one strum. Gives the effect of playing a chord and then bending all the notes in the chord up several steps to a new chord.
In fact, in the above example, you can strum the full six string G chord once - and without strumming again, hammer the new chord shape on the D G and B strings in first position and slide the shape up to the XX453X position and effectively slide from a G to a that D7 voicing all in one strum - even though the two chords have no fretted strings in common.
If you don't do it as I described, you will have to strum or otherwise play the new chord after you slide up to it - before it will be heard as the new chord. If that is acceptable in your arrangement - it may be easier just to form a part of the new chord (perhaps the root and 3rd - depending on voicing chosen) and slide the partial new chord shape up in the manner described and then strum again after your target chord is reached - to establish the new chord.
If you want to slide from a chord played higher up the fret board to another chord in a higher position, and they don't share any commonality between the two chord shapes, the easiest thing to do is exactly what Dr. Mayhem suggested - form the new chord in one position and slide it up.
For example, in a song I play often - I go from a 5th fret A Barre chord (using E shaped Barre chord formation) to an E7 played around the 7th fret as 076707 (B7 shape). To give the effect of sliding up to that chord, I form it on the 6th fret 065606 and slide it up one fret. There is no other way to slide from the Barre Chord to the more open chord without doing it this way.
The foregoing illustrates that the exact technique to slide between two chords with your chosen voicing - will vary, depending on the way each chord is formed. Most likely you will want to develop different methods for each of the various chord transitions.
Once you get used to a certain slide from one chord to another, it becomes second nature. It takes practice in the beginning, just like learning a new chord shape that you have never tried. Once you master it, you can throw it in wherever and whenever it fits - to spice up any guitar arrangement.