We study an online variant of the celebrated housing market problem, where each agent owns a single house and seeks to exchange it based on her preferences. In this online setting, agents may arrive and depart at any time, meaning not all agents are present in the housing market simultaneously. We extend the well-known serial dictatorship and top trading cycle mechanisms to the online scenario, aiming to retain their desirable properties, such as Pareto efficiency, individual rationality, and strategy-proofness. These extensions also seek to prevent agents from strategically delaying their arrivals or advancing their departures. We demonstrate that achieving all these properties simultaneously is impossible and present several variants that achieve different subsets of these properties.