  We study a family of symmetric functions $\hat F_z$ indexed by involutions
$z$ in the affine symmetric group. These power series are analogues of Lam's
affine Stanley symmetric functions and generalizations of the involution
Stanley symmetric functions introduced by Hamaker, Pawlowski, and the first
author. Our main result is to prove a transition formula for $\hat F_z$ which
can be used to define an affine involution analogue of the
Lascoux-Sch\"utzenberger tree. Our proof of this formula relies on Lam and
Shimozono's transition formula for affine Stanley symmetric functions and some
new technical properties of the strong Bruhat order on affine permutations.
