These closures can be found, for example, here for codim 2 continuation and here for periodic orbits where they are used to update the section at every continuation steps.
These closure can be removed using theupdate! method dispatch on a BifurcationProblem. Their removal would greatly simplify the code. Closures are indeed notarially difficult to reason with.
Some of them can also be removed using dispatch on the record_from_solution.