  Given a smooth projective algebraic surface X, a point O in X and a big
divisor D on X, we consider the set of all Newton-Okounkov bodies of D with
respect to valuations of the field of rational functions of X centred at O, or,
equivalently, with respect to a flag (E,p) which is infinitely near to O, in
the sense that there is a sequence of blowups mapping the smooth, irreducible
rational curve E to O. The main objective of this paper is to start a
systematic study of the variation of these infinitesimal Newton-Okounkov bodies
as (E, p) varies, focusing on the case X = P2.
