  We prove that the diametral diameter two properties are inherited by
$F$-ideals (e.g., $M$-ideals). On the other hand, these properties are lifted
from an $M$-ideal to the superspace under strong geometric assumptions. We also
show that all of the diametral diameter two properties are stable under the
formation of corresponding K\"othe-Bochner spaces (e.g., $L_p$-Bochner spaces).
Finally, we investigate when the projective tensor product of two Banach spaces
has some diametral diameter two property.
