We study a generalized class of nonlocal evolution equations which includes those arising in the modelling of electrified film flow down an inclined plane, with applications in enhanced heat or mass transfer through interfacial turbulence. Global existence and uniqueness results are proved and refined estimates of the radius of the absorbing ball in $L^2$ are obtained in terms of the parameters of the equations (the length of the system and the dimensionless electric field-measuring parameter multiplying the nonlocal term). The established estimates are compared with numerical solutions of the equations which in turn suggest an optimal upper bound for the radius of the absorbing ball. A scaling argument is used to explain this and a general conjecture is made based on extensive computations.