I am having trouble getting this question framed properly. When I am building a solo, say an A minor blues solo (A minor pentatonic), do I play in the A scale over the whole progression or do I switch to a D scale then an E scale as the chords change?
You've identified two ways of choosing scales for a chord progressions there.
One way is to use a single, root scale throughout the entire progression. In blues, you can do this because the minor pentatonic scale works over the 3 chord (I IV V) blues progressions.
For example, A minor pentatonic (or more likely the A blues scale) would typically be used over a blues progression of A7 (I), D7 (IV) and E7 (V).
Another example would be to use the C major scale over a progression of C / Dm / F / G because all these chords are built from notes of the C major scale.
Spend a few minutes watching the video below for more on this "chord scale" concept.
Being able to identify when a single root scale can be used over a sequence of related chords is a crucial step forward in developing as a soloist/lead guitarist.
The other way of using scales is to use a separate scale for each chord in the progression. This is (confusingly, given the nature of the 1st way) often referred to as the chord-scale system.
Basically, this is where you connect each chord with a number of scale choices.
In most cases, if you can apply one root scale to an entire progression, it will sound less disjointed than if you tried to change scales with the chord. Often the flow of harmony through a progression will sound more "natural" if you play that root scale throughout.
However, be aware that some progressions are what we call non-diatonic, meaning the chords in the progression are not all related to one scale. This is where you have to choose more than one scale, connecting each chord to a related scale.