Suppose we have following commutative diagram (not a square) of schemes:

$X\xrightarrow{p_1} Y$, $Y\xrightarrow{\pi_2} Z$, $X\xrightarrow{\pi_1} W$, $W\xrightarrow{p_2} Z$.

Let $E$ be a coherent sheaf on $X$.

Is there a natural morphism $R^ip_{1*}E\rightarrow \pi_2^*R^ip_{2*}(\pi_{1*}E)$ for all i?

or

Is there a natural morphism $\pi_2^*R^ip_{2*}(\pi_{1*}E)\rightarrow R^ip_{1*}E$ for all i?

Suppose we have following commutative diagram (not a square i.e not a base change) of schemes:

$X\xrightarrow{p_1} Y$, $Y\xrightarrow{\pi_2} Z$, $X\xrightarrow{\pi_1} W$, $W\xrightarrow{p_2} Z$.

Let $E$ be a coherent sheaf on $X$.

Is there a natural morphism $R^ip_{1*}E\rightarrow \pi_2^*R^ip_{2*}(\pi_{1*}E)$ for all i?

or

Is there a natural morphism $\pi_2^*R^ip_{2*}(\pi_{1*}E)\rightarrow R^ip_{1*}E$ for all i?

Suppose we have following commutative diagram (not a square) of schemes:

$X\xrightarrow{p_1} Y$, $Y\xrightarrow{\pi_2} Z$, $X\xrightarrow{\pi_1} W$, $W\xrightarrow{p_2} Z$.

Let $E$ be a coherent sheaf on $X$.

Is there a natural morphism $R^ip_{1*}E\rightarrow \pi_2^*R^ip_{2*}(\pi_{1*}E)$ for all i?

Suppose we have following commutative diagram (not a square) of schemes:

$X\xrightarrow{p_1} Y$, $Y\xrightarrow{\pi_2} Z$, $X\xrightarrow{\pi_1} W$, $W\xrightarrow{p_2} Z$.

Let $E$ be a coherent sheaf on $X$.

Is there a natural morphism $R^ip_{1*}E\rightarrow \pi_2^*R^ip_{2*}(\pi_{1*}E)$ for all i?

or

Is there a natural morphism $\pi_2^*R^ip_{2*}(\pi_{1*}E)\rightarrow R^ip_{1*}E$ for all i?