Following on from looking at the major triads in open D, a next logical step is to look at minor triads.
Getting from minor triad to a major triad is fairly simple. We just need to move the major 3rds down a fret. From looking at an open D equivalent of CAGED, we would expect there to be 3 basic shapes for our minor triads, the same as major triads.
One little wrinkle however is that our first shape is all the open strings. This means that we cannot just move the major third down to a minor third. To get round this we will start by playing the D shape at 12th fret. Since we are really interested in shapes we can move around the fretboard, this is fine as the shape will work for all other chords. We just can't use it to play a D minor using open strings. There is another solution for this we will come back to at the end.
So if we take the 3 chord shapes from the end of the CAGED in open D post and flatten the third we get the following 3 shapes.
One other small difference is that on the Bm shape, I typically play the 1st string whereas I don't on the B major shape. This is just because the major third is in a awkward position to finger where as the minor third works much more easily both as an open chord and when you move the shape around.
For the Dm shape, the notes I play vary depending on where I am on the context hence why I have left them all marked in. Probably the most common version I would play would be to omit the 5th and 6th strings and just play the top 4 strings. Next most common would be to omit the 4th and 5th strings.
Other than this, all the usual rules discussed when talking about triads in open D apply. So 3 adjacent strings (excluding the 6th string) can form a minor triad and you can then shift between the inversions of that triad by moving across string sets or up and down shapes.
Finally, at the start I mentioned an alternative shape for playing a Dm using the open strings. We first need to include the minor third. The easiest place is 3rd fret on any of the 1st, 4th and 6th strings. I'd normally go for the 4th unless there was a compelling to use one of the others. However since we also have the major third as the open 3rd string, we need to play something else on that string. Fortunately we can play the 5th using the 3rd fret on the 3rd string. So my go-to Dm shape using the open strings is as follows