Let G be a graph with n vertices and m edges. The edge corona \(G \diamond H\) of two graphs G and H is constructed by taking one copy of G and m copies of H and making the end vertices of the \(j{\text {th}}\) edge of G adjacent to every vertex in the \(j{\text {th}}\) copy of H, for \(j = 1, \ldots ,m\) . Let \(\mathscr {F}\) represent the family of graphs that includes complete graphs with at least 2 vertices and graphs of diameter 2, in which the diametrically opposite vertices of any peripheral vertex are non-adjacent. In this paper, we describe the distance spectrum of \(G \diamond H\) using the adjacency eigenvalues of G and H, when both G and H are regular and \(G \in \mathscr {F}\) . Several constructions are proposed using line graphs, iterated line graphs, double graphs, strong double graphs, and complement graphs to obtain infinitely many distance non-cospectral pairs of distance equienergetic graphs and non-isomorphic pairs of distance cospectral graphs. Additionally, we derive the distance Laplacian spectrum of \(G \diamond H\) based on the distance Laplacian spectrum of G and Laplacian spectrum of H, under the condition that G is transmission regular and belongs to \(\mathscr {F}\) , while H is arbitrary. Further, we find the distance signless Laplacian spectrum of \(G \diamond H\) based on the distance signless Laplacian spectrum of G and signless Laplacian spectrum of H, when G is transmission regular and belongs to \(\mathscr {F}\) and H is regular. Moreover, we construct infinitely many non-isomorphic pairs of distance Laplacian cospectral graphs and distance signless Laplacian cospectral graphs.