On-stack replacement (OSR) dynamically transfers execution between different code versions. This mechanism is used in mainstream runtime systems to support adaptive and speculative optimizations by running code tailored to provide the best expected performance for the actual workload. Current approaches either restrict the program points where OSR can be fired or require complex optimization-specific operations to realign the program's state during a transition. The engineering effort to implement OSR and the lack of abstractions make it rarely accessible to the research community, leaving fundamental question regarding its flexibility largely unexplored. In this article we make a first step towards a provably sound abstract framework for OSR. We show that compiler optimizations can be made OSR-aware in isolation, and then safely composed. We identify a class of transformations, which we call live-variable equivalent (LVE), that captures a natural property of fundamental compiler optimizations, and devise an algorithm to automatically generate the OSR machinery required for an LVE transition at arbitrary program locations. We present an implementation of our ideas in LLVM and evaluate it against prominent benchmarks, showing that bidirectional OSR transitions are possible almost everywhere in the code in the presence of common, unhindered global optimizations. We then discuss the end-to-end utility of our techniques in source-level debugging of optimized code, showing how our algorithms can provide novel building blocks for debuggers for both executables and managed runtimes.