  We consider Ricci flow invariant cones C in the space of curvature operators
lying between nonnegative Ricci curvature and nonnegative curvature operator.
Assuming some mild control on the scalar curvature of the Ricci flow, we show
that if a solution to Ricci flow has its curvature operator which satsisfies
R+\epsilon I \in C at the initial time, then it satisfies R +K\epsilon I \in C
on some time interval depending only on the scalar curvature control. This
allows us to link Gromov-Hausdorff convergence and Ricci flow convergence when
the limit is smooth and R + I \in C along the sequence of initial conditions.
Another application is a stability result for manifolds whose curvature
operator is almost in C. Finally, we study the case where C is contained in the
cone of operators whose sectional curvature is nonnegative. This allow us to
weaken the assumptions of the previously mentioned applications. In particular,
we construct a Ricci flow for a class of (not too) singular Alexandrov spaces.
