Do you mean this: Perturbation parameter for the static step ?
It’s not that useful to be honest because it forces full linearity of a static step and subsequent steps are independend of each other. In Abaqus, it’s used mostly for linear load cases.
This sounds more like application for import functionality in Abaqus (transferring the deformed mesh with material state between the analyses). In CalculiX, you can only manually reuse the deformed meshes.
However, with multiple nonlinear general static steps (where the state of the model from the end of one step is used as the base state in the second step), boundary conditions and loads that can be activated/deactivated within each step and controlled by amplitudes, this should be doable. Especially since you can even activate and deactivate elements and contact pairs with the custom *MODEL CHANGE keyword.
Indeed, force control often leads to non-convergence of nonlinear analyses. So it’s advised to use displacement control instead whenever possible (you can always adjust it based on the reaction forces or follow it with force control step). For preload as well.