First asymptotic relations of Voronovskaya-type for rational operators of Shepard-type are shown. A positive answer in some senses to a problem on the pointwise approximation power of linear operators on equidistant nodes posed by Gavrea, Gonska and Kacsó is given. Direct and converse results, computational aspects and Grüss-type inequalities are also proved. Finally an application to images compression is discussed, showing the outperformance of such operators in some senses.