Here is my decision. It looks like previous one, but it’s not player himself who moves, but box created in it.
Anyone knows is this problem has been resolved yet?
I am encountering it now and it would be nice to know if they added some kind of option to fix this 
It works! Good solution, thx!
2021 and apparently no fix yet
Sending a message back to past of May 2014, from June 2021:
This hasn’t been fixed yet 
Cant believe this is still an issue. God ■■■■.
What do you mean by “On edge of door opposite its hinges”?
Here is a simple and short solution to this problem. Simply update character movement component at every frame in Tick function:
FHitResult OutHit;
GetCharacterMovement()->SafeMoveUpdatedComponent(FVector(0.f, 0.f, 0.01f), GetActorRotation(), true, OutHit);
GetCharacterMovement()->SafeMoveUpdatedComponent(FVector(0.f, 0.f, -0.01f), GetActorRotation(), true, OutHit);
Credit for this solution goes to people in this post: character collision not working when stood still, works when moving - #6 by Ceteri
I tried that but it didn’t work. Actually, it partially worked, but not a 100%. Sometimes moving object still goes through character, other times it pushes character as expected. I don’t know why it still happens sometimes.