Thanks, DYKWIA for your answer.

What do you mean with "can't walk between T1 & T3 these days"?
There is a walking bridge between these terminals and I read several comments about no problems to walk there.
Also, Dubai Airport Website says: "Terminal 1’s D-Gates are reachable via a short trip on an airport train. In Terminal 3, passengers can walk between B-Gates and C-Gates and take a short airport train to A-Gates."
Did they change something?

I did some research on the terminal maps and forum threads.
So in T3 Arrival, there should be a baggage security check and after this, you can go upstairs to T3 Departure or to the Transfer Desk for transfer to T2. I don't see any boarding pass check there?
So getting to T3 Departures should be no problem.

Walking from T3 to T1 is should be able as described above?

Now, as you said, the problem is to get back to transit for Terminal 2. As there is no walkway, I need to get to the transit bus.
I read one comment from 2017 that you can go to the "transfer desk K on Terminal 3 concourse B". They can arrange a shuttle to Terminal 2.

Can someone confirm this?