  In this paper, we introduce the flag-wise positively curved condition for
Finsler spaces (the (FP) Condition), which means that in each tangent plane, we
can find a flag pole in this plane such that the corresponding flag has
positive flag curvature. Applying the Killing navigation technique, we find a
list of compact coset spaces admitting non-negatively curved homogeneous
Finsler metrics satisfying the (FP) Condition. Using a crucial technique we
developed previously, we prove that most of these coset spaces cannot be
endowed with positively curved homogeneous Finsler metrics. We also prove that
any Lie group whose Lie algebra is a rank $2$ non-Abelian compact Lie algebra
admits a left invariant Finsler metric satisfying the (FP) condition. As
by-products, we find the first example of non-compact coset space $S^3\times
\mathbb{R}$ which admits homogeneous flag-wise positively curved Finsler
metrics. Moreover, we find some non-negatively curved Finsler metrics on
$S^2\times S^3$ and $S^6\times S^7$ which satisfy the (FP) condition, as well
as some flag-wise positively curved Finsler metrics on $S^3\times S^3$,
shedding some light on the long standing general Hopf conjecture.
