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Let \( G \) be an edge-colored connected graph. A path \( P \) is a proper path in \( G \) if no two adjacent edges of \( P \) are colored the same. An edge coloring is a proper-path coloring of \( G \) if every pair \( u, v \) of distinct vertices of \( G \) is connected by a proper \( u-v \) path in \( G \). The minimum number of colors required for a proper-path coloring of \( G \) is the proper connection number \( \text{pc}(G) \) of \( G \). We study proper-path colorings in those graphs obtained by some well-known graph operations, namely line graphs, powers of graphs, coronas of graphs, and vertex or edge deletions. Proper connection numbers are determined for all iterated line graphs and powers of a given connected graph. For a connected graph \( G \), sharp lower and upper bounds are established for the proper connection number of (i) the \( k \)-iterated corona of \( G \) in terms of \( \text{pc}(G) \) and \( k \), and (ii) the vertex or edge deletion graphs \( G-v \) and \( G-e \), where \( v \) is a non-cut-vertex of \( G \) and \( e \) is a non-bridge of \( G \), in terms of \( \text{pc}(G) \) and the degree of \( v \). Other results and open questions are also presented.