In recent papers [8], [9] the technique for a local and global errors estimation and the local-global step size control were presented to solve ordinary differential equations by means of variable-coefficients multistep methods with the aim to attain automatically any reasonable accuracy set by the user for the numerical solution. Here, we extend those results to the class of multistep formulas with fixed coefficients implemented on nonuniform grids. We give a short theoretical background and numerical examples which clearly show that the local-global step size control works in multistep methods with polynomial interpolation of the numerical solution when the error of interpolation is sufficiently small.