The containers used for steel slag transportation to the recycling depot undergo high-temperature gradients and often deform plastically. Also, parts of the thereby solidified slag adhere to the pot walls, causing demolding problems and wear. A thorough finite-element analysis of the heat transfer, initiated by filling the pots and essentially driven by radiation, and the thermal stresses is performed. Due to periodic fill-in and discharge, these are assumed to admit a quasi-stationary state referring to the pot temperature before their emptying. The so obtained results aid optimizing the pot shape and the transport process in terms of minimizing maintenance and anti-adhesive cladding. A layer of solidified slag is found to exist throughout the transport process. Although the phase change occurs almost instantaneously at a liquid-solid interface, the "mushy zone" is considered in the accompanying analytical study of the associated Stefan-type problem. Good correlations of the predicted temperatures to the measured ones and the resultant surface damaging are obtained.