When a public road has or shall have been improved under the provisions of this article and in the course of the improvement the grade of the roadbed shall have been or shall be changed so that it shall not conform to the grade of the remaining portion of the road, including the sidewalks, the authorities of each municipality through which the road passes may cause the remaining portion of the road within the limits of the municipality to be graded and formed so as to conform in grade to that established for the roadway constructed under the provisions of this article, and may cause the work to be done under the proceedings and in the manner provided by law in their several municipalities for the grading of the streets thereof.