Post shave cleaning

Warm rinse of brush under tap; same with razor and blade; gently dry brush with towel and leave to air dry; dry individual razor components (i.e for a 3 piece DE) or whole razor with towel. Leave blade out of razor til next time.
 
Warm rinse of brush under tap; same with razor and blade; gently dry brush with towel and leave to air dry; dry individual razor components (i.e for a 3 piece DE) or whole razor with towel. Leave blade out of razor til next time.
I do exactly the same, the only additional things i do is use the remaining lather as a final face wash, before my cold water rinse.
 
The wear I would do on the razor thread constantly opening and closing it for cleaning outweights any potential damage I might cause to the finish with soap scum. Ocassional soak in white vinegar solves any residues issues. Damned vax glue buildup needs bit more elbow grease to get rid off.

I wash natural hair brushes with shampoo/conditioner after every shave. Keeps them soft and pliable the way I like them.