A non-variational approach to the construction of new 'higher-order' conservation laws of the family of nonlinear equations α(ut + 3uux) + β(utxx + 2uxuxx + uuxxx) − γuxxx = 0

► Multiplier and symmetry approach to construct conservation laws for a class of combined Camassa-Holm, Hunter Saxton and Inviscid Burgers equations are studied. ► Higher-order multipliers are first determined and then the new conserved flows are constructed. ► Relationship between the flows and symmetries are identified by the relationship between the multipliers and symmetries.