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We study regular coverings of graphs and manifolds with a focus on propertiesof the heat equation. In particular, we look at stochastic incompleteness, theFeller property and uniform transience; and investigate the connection betweenthe validity of these properties on the base space and its covering. For bothgraphs and manifolds, we prove the equivalence of stochastic incompleteness ofthe base and that of its cover. Along the way we also give some new conditionsfor the Feller property to hold on graphs.

Coverings and the heat equation on graphs: stochastic incompleteness, the Feller property and uniform transience