We consider induced coactions on C*-algebras along a homomorphism of locally compact quantum groups which need not give a closed quantum subgroup. Our approach generalizes the induced coactions constructed by Vaes, and also includes certain fixed point algebras. We focus on the case when the homomorphism satisfies a quantum analogue of properness. Induced coactions along such a homomorphism still admit the natural formulations of various properties known in the case of a closed quantum subgroup, such as imprimitivity and adjointness with restriction. Also, we show a relationship of induced coactions and restriction which is analogous to base change formula for modules over algebras. As an application, we give an example that shows several kinds of 1-categories of coactions with forgetful functors cannot recover the original quantum group. Contents 1. Introduction 1 2. Preliminaries 3 3. Formal aspects of induced coactions 6 4. Induced coactions along a proper homomorphism 10 5. Analogue of base change 16 6. Morita and K-theoretic properties 21 7. Example: double crossed products with a discrete abelian group 25 Appendix A. Constructions for locally compact quantum groups and coactions 28 References 32