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We investigate transport equations associated to a Lipschitz field on somesubspace of $\mathbb{R}^N$ endowedwith a general measure $\mu$ in$L^{p}$-spaces $1 < p <\infty$, extending the results obtained in two previouscontributions of the author in the $L^{1}$-context. We notably prove thewell-posedness of boundary-value transport problems with a large variety ofboundary conditions. New explicit formula for the transport semigroup are inparticular given.

An $L^{p}$--approach to the well-posedness of transport equations associated to a regular field