At last - I've found a solution to my problem! I've noted the steps below if anyone needs. Thank you again @tyflow for your support throughout the thread! 
https://streamable.com/p5ysw6
1. Animate the door using soft selection
2. Create a plane with roughly same topology as the door, in front of the door to act as its cover
3. Skin wrap the plane to the animated mesh
4. Point cache the plane and hide it
5. Create a tyIcon, align to the hidden plane
6. Create a tyFlow object with the following nodes: Birth (0;0;1), Shape, Position icon, Display (geometry)
7. Shape: set to reference the point cached plane, scroll down and enable 'animated geometry' and choose your frame range
8. Position icon: set location to pivot (in my case that matches)
https://streamable.com/p5ysw6
1. Animate the door using soft selection
2. Create a plane with roughly same topology as the door, in front of the door to act as its cover
3. Skin wrap the plane to the animated mesh
4. Point cache the plane and hide it
5. Create a tyIcon, align to the hidden plane
6. Create a tyFlow object with the following nodes: Birth (0;0;1), Shape, Position icon, Display (geometry)
7. Shape: set to reference the point cached plane, scroll down and enable 'animated geometry' and choose your frame range
8. Position icon: set location to pivot (in my case that matches)