We study four asymptotic smoothness properties of Banach spaces, denoted
T
p
,
A
p
,
N
p
\mathsf {T}_p,\mathsf {A}_p, \mathsf {N}_p
, and
P
p
\mathsf {P}_p
. We complete their description by proving the missing renorming characterization for
A
p
\mathsf {A}_p
. We show that asymptotic uniform flattenability (property
T
∞
\mathsf {T}_\infty
) and summable Szlenk index (property
A
∞
\mathsf {A}_\infty
) are three-space properties. Combined with the positive results of the first-named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three-space problem for this family of properties. We also derive from our characterizations of
A
p
\mathsf {A}_p
and
N
p
\mathsf {N}_p
in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes.