What can cause disc brakes not to react after keeping/moving the bike upright, then recover? Recently, I experienced a strange situation with the Tektro hydro disc brakes on my hybrid bike (would the model be helpful here?):

I carried it upstairs at a friend's place, and later moved it downstairs by holding it upright and slowly pushing it downwards. When I started riding home, I noticed that the rear brake almost didn't react to any pull, I was able to pull the lever with minimum resistance until it almost hit the bar. I felt almost(?) no braking action. A few minutes into the ride, the brake fully recovered to the normal amount of resistance/braking force.

What happened here? What might cause such a behaviour? Could it be the bumpy ride down the stairs?

I never experienced fading on that brake so far, from the description it sounds very similar, though the brake was completely cool at the start of the ride.
As always, it's hard to diagnose something with such a brief description and without looking at it, but the most likely cause is air in your brake system.

Brake systems are designed to work with your bike standing still, where any small amount of air is pushed by gravity to the reservoir of the levers, keeping the system working well.

When you move your bike around and put it in strange positions, or when you store it vertically, this changes, gravity now pushes oil into the reservoir and air to the lever. This means when you start riding there is actual air somewhere where it shouldn't be and the brake don't feel right. If you keep your bike still for a few minutes this "resets", and air goes back to the reservoir and the brakes star working ok.

On the beginning of a ride, and mostly if you store your bike vertically, you should always test the brakes and give them a few minutes for oil to go "down" and air "up", and press the levers a few times because this creates mechanical pressure on the lever so that it gets filled with oil and air is pushed into the reservoir. This is actual advised in some brakes manuals.

It doesn't hurt to bleed your brakes and take as much air out as you can, and now that you are at it, check if the system is well isolated, i.e., check for small spills of oil in the calipers and levers and around the tubes. A problem in the system is not a likely cause as the brakes would always perform poorly instead of only after you moved your bike, but it doesn't hurt to take a look.

Happy riding!