The phrase "to move the mass" means to physically or emotionally influence a large group of people or objects to take action or make changes. Full definition
I regards to the weight (mass, really) distribution, a problem with moving the mass to benefit weight transfer on braking is that it reduces stability. (mechanics.stackexchange.com)
The calculation involves working out the difference in gravitational potential — a measure of the work done by gravity in moving a mass from one location to another — between the centre and surface. (newscientist.com)
We believe that to help move the mass market towards sustainability, momentum needs to be gathered around products that are significantly better than their counterparts yet are not all perfectly sustainable. (treehugger.com)