Pop in E (suspended) — alternating bass fingerstyle
The note that tells Esus4 apart from E, like the one that tells Asus2 apart from A, always sits on a middle or treble string, never on the bass strings. The alternating bass pattern the thumb plays therefore stays exactly the same across all four chords: contrary to what you might expect, the suspension doesn't complicate the right hand's job at all — it only concerns the fingers playing above it.
Tips
First learn the thumb motion alone, on a single held chord: it won't change when you move to the other three.
Let the fingers playing the suspended note move independently, without the thumb slowing down or speeding up.
If the thumb stumbles during the Esus4→E or Asus2→A resolution, the problem is in the fingers above, not the thumb itself.
Chord progression — E
Bar 1
Esus4
Bar 2
E
Bar 3
Asus2
Bar 4
A