Hydrodynamic limits of particle systems with selection are related to a class of free boundary problems (FBP) associated with second order parabolic PDE. This relation has been established rigorously for some selection models but remains open for others, where the difficulty lies in questions of regularity of the free boundary. We introduce a weak formulation that does not involve the notion of a free boundary, but reduces to a FBP when classical solutions exist. For the injection-branching-selection model of Brownian particles on the line with arbitrarily varying removal rates, the approach yields the characterization of limits in terms of a PDE beyond cases where classical solutions are known (or even expected) to exist. Results on a higher dimensional model of motionless branching particles will also be described, as well as a formal relation to the Stefan FBP and open problems.