  We study the dynamics of the group of isometries of $L_p$-spaces. In
particular, we study the canonical actions of these groups on the space of
$\delta$-isometric embeddings of finite dimensional subspaces of $L_p(0,1)$
into itself, and we show that for $p \neq 4,6,8,\ldots$ they are
$\varepsilon$-transitive provided that $\delta$ is small enough. We achieve
this by extending the classical equimeasurability principle of Plotkin and
Rudin. We define the central notion of a Fra\"iss\'e Banach space which
underlies these results and of which the known separable examples are the
spaces $L_p(0,1)$, $p \neq 4,6,8,\ldots$ and the Gurarij space. We also give a
proof of the Ramsey property of the classes $\{\ell_p^n\}_n$, $p\neq 2,\infty$,
viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an
arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well
as to the notion of a spreading vector of Matou\v{s}ek and R\"{o}dl. Finally,
we give a version of the Kechris-Pestov-Todorcevic correspondence that links
the dynamics of the group of isometries of an approximately ultrahomogeneous
space $X$ with a Ramsey property of the collection of finite dimensional
subspaces of $X$.
