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We study $2d$ and $3d$ gravity theories on spacetimes with causal (timelike or null) codimension one boundaries while allowing for variations in the position of the boundary. We construct the corresponding solution phase space and specify boundary degrees freedom by analysing boundary (surface) charges labelling them. We discuss $Y$ and $W$ freedoms and change of slicing in the solution space. For $D$ dimensional case we find $D+1$ surface charges, which are generic functions over the causal boundary. We show that there exist solution space slicings in which the charges are integrable. For the $3d$ case there exists an integrable slicing where charge algebra takes the form of Heisenberg $\oplus\ {\cal A}_3$ where ${\cal A}_3$ is two copies of Virasoro at Brown-Henneaux central charge for AdS$_3$ gravity and BMS$_3$ for the $3d$ flat space gravity.
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