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We assess the prospects for algorithms within the general framework of quantum annealing (QA) to achieve a quantum speedup relative to classical state of the art methods in combinatorial optimization and related sampling tasks. We argue for continued exploration and interest in the QA framework on the basis that improved coherence times and control capabilities will enable the near-term exploration of several heuristic quantum optimization algorithms that have been introduced in the literature. These continuous-time Hamiltonian computation algorithms rely on control protocols that are more advanced than those in traditional ground-state QA, while still being considerably simpler than those used in gate-model implementations. The inclusion of coherent diabatic transitions to excited states results in a generalization called diabatic quantum annealing (DQA), which we argue for as the most promising route to quantum enhancement within this framework. Other promising variants of traditional QA include reverse annealing and continuous-time quantum walks, as well as analog analogues of parameterized quantum circuit ansatzes for machine learning. Most of these algorithms have no known (or likely to be discovered) efficient classical simulations, and in many cases have promising (but limited) early signs for the possibility of quantum speedups, making them worthy of further investigation with quantum hardware in the intermediate-scale regime. We argue that all of these protocols can be explored in a state-of-the-art manner by embracing the full range of novel out-of-equilibrium quantum dynamics generated by time-dependent effective transverse-field Ising Hamiltonians that can be natively implemented by, e.g., inductively-coupled flux qubits, both existing and projected at application scale.