I don't know what material you have, or what size and weight the pieces are.
I accumulate stuff in a builders bag. If I had to carry rubble down steps, I would carry two buckets at a time, for balance, and wear flexible shoes with rubber soles.
If it's light scrap wood, I use a polypropylene garden bag. I use a shredder for prunings.
In my case, a builders bag fits in the back of my vehicle, so I put in in empty, then fill it up when going to the tip.
In my case, a builders bag fits in the back of my vehicle, so I put it in empty, then use
rubbish removal Richmond when it’s full and ready to be taken away.