Soft solids such as gels and elastomers can be compliant enough to deform when a droplet lies on their surface, in particular at the line of contact between the solid, the liquid and the atmosphere. While axisymmetric contact line motion has received a lot of attention, much less is known about droplets moving on soft substrates, a configuration often encountered in applications in which symmetry may be lost. We investigate here the dynamic properties of droplets sliding on thick viscoelastic layers. We show that the partition of energy dissipation between the liquid and the solid sets the shape and velocity of droplets. When dissipation parts equally between the liquid and the solid, droplet dynamics are similar to that of droplets on rigid substrates. In the opposite case, we observe shapes that indicate the presence of contact angle hysteresis. We compare our observations to a non-linear model of the wetting of soft solids that we proposed recently. We find the model to be in excellent agreement with our data, in particular regarding the prediction of the hysteresis that we show to be only apparent. This work opens fondamental questions on the connection between the properties of the substrate and the dynamics, shapes and fragmentation of moving droplets that are relevant to all applications where soft gel coatings may be used.