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We define the structure of spaces with labelled partitions which generalizesthe structure of spaces with measured walls and study the link between actionsby automorphisms on spaces with labelled partitions and isometric affineactions on Banach spaces, and more particularly, on $L^p$ spaces. We buildnatural spaces with labelled partitions for the action of various contructionsof groups, namely: direct sum; semi-direct product; wreath product andamalgamated free product. We apply this to prove that the wreath product of agroup with property $PL^p$ by a group with Haagerup property has property$PL^p$ and the amalgamated free product over finite subgroups of groups withproperty $PL^p$ has property $PL^p$.

Spaces with labelled partitions and isometric affine actions on Banach spaces