Let $X$ be a smooth projective variery. Let $-K_X$ be nef , then which type of Canonical metric In the sense of Einstein type metric is suitable for it.
In fact when $K_X$ or $-K_X$ is ample WE know that WE have $ Ric(\omega)=-\omega$, and $ Ric(\omega)=\omega$, when X is K-stable.
In my opinion when $-K_X$ be nef then WE know that the Albanese map is surjective, I.e $\pi:X\to Alb(X)$ is surjective and the best Canonical metric is the relative Kahler Einstein metric along Albanese map $$Ric_{X/S}(\omega)=-\xi(s)\omega$$ As soon as the relative tangent sheaf $T_{X/S}$ is stable In the sense of Mumford.
Where here $S=Alb(X)$ is Albanese variety and $\omega$ is the relative Kahler form and $\xi(s)$ is fiberwise constant.
In this case the right flow is the following Hyperbolic Relative Kahler Ricci flow.
$$\frac{\partial^2\omega}{\partial s'\partial t}=-Ric_{X/S}\omega(s',t)-\xi(s)\omega_{s'}(t)$$
Where here $s'=\frac{1}{s}$ and $s\to 0$
See here the definition of relative Kahler form see my answer to this post What is a Kählerian variety?
Also about Albanese map see my answer to this post What is the Albanese map good for?
For nef Line bundle see my answer to this post nef Line bundles over Kähler manifolds
When Tangent sheaf of a Fano Variery is nef see JPD paper https://arxiv.org/abs/1712.03725
My motivation is that let $\pi:X\to \Delta$ be a surjective holomorphic fibre space from a projective variery to a holomorphic disc.
Let $K_{X_t}$ is ample then the Canonical divisor $K_{X_0}$ is nef and finding a type of Canonical metric on central fiber is an open question and it may admits several différent type of Canonical metrics including twisted KE
See my post about Beauville-Bogomolov decomposition by using relative Kahler Ricci flow along Albanese map https://mathoverflow.net/questions/277561/kähler-ricci-flow-approach-for-beauville-bogomolov-type-decomposition
Hassan Jolany