Multidimensional transition fronts for Fisher-KPP reactions

We study entire solutions to homogeneous reaction-diffusion equations in several dimensions with Fisher-KPP reactions. Any entire solution $0<u<1$ is known to satisfy $$\lim_{t\to -\infty} \sup_{|x|\le c|t|} u(t,x) = 0 \qquad \text{for each $c<2\sqrt{f'(0)}\,$,}$$ and we consider here those satisfying $$\lim_{t\to -\infty} \sup_{|x|\le c|t|} u(t,x) = 0 \qquad \text{for some $c>2\sqrt{f'(0)}\,$.}$$ When $f$ is $C^2$ and concave, our main result provides an almost complete characterization of transition fronts as well as transition solutions with bounded width within this class of solutions.