This sounds like a job for the Snap tool which appears to require a Standard (ArcEditor) or Advanced (ArcInfo) license.
"One use case for this tool is to rectify the differences in shared or common boundaries between two datasets by snapping the vertices in one boundary to the vertices, edges, or end points of the other. If the input features do not have enough vertices to match the exact curvature of the other boundary, vertices can be added to the input features using the Densify tool to allow for an added level of detail."
An alternative may be to use the Linear Referencing approach suggested by @simplexio - perhaps the Overlay Route Events tool with input points and output lines. However, I would try Densify/Snap first, if you have the required licensing.