We have developed a global approach to define bisimulation distances which goes somehow further away than the bisimulation distances based on the bisimulation game, previously proposed by some other authors. Our proposal is based on the cost of transformations: how much we need to modify one of the compared processes to obtain the other. Our original definition only covered finite processes, but a coinductive approach extends it to cover infinite but finitary trees. We have shown many interesting properties of our distances, and we wanted to prove their continuity with respect to projections, bur unfortunately we have not been able to accomplish that task. However, we have obtained several partial results that we now present in this paper.