I'm trying to work out a result in Fulton's intersection theory and I think I need the following basic result about base change of projective cones (whose support may not be the entire base scheme).

Let $X$ be a scheme of finite type over a field, and $S^\cdot$ a sheaf of graded $\mathcal{O}_X$ algebras on $X$ generated by $S^1$ over $S^0$, with the natural map $\mathcal{O}_X \rightarrow S^0$ surjective. $P = \textrm{Proj}(S^\cdot)$, along with it's nautral projection $\pi$ to $X$, is a projective cone on $X$ with support defined by the ideal sheaf $\ker( \mathcal{O}_X \rightarrow S^0)$.

Let $i: Z \rightarrow X$ be a closed immersion. Base change $\pi$ along $i$. Then, do we have:

1.) The fiber product is the cone corresponding to the pullback $i^*(S^\cdot)$ sheaf of algebras on $Z$.

2.) The induced morphism on cones pulls back $O_P(1)$ to $O_Z(1)$ where these sheaves denote the canonical sheaves on the projective cones over $X$ and $Z$ respectively.

In his intersection theory book, Fulton only states (1,2) as true in a more restrictive setting ($i$ can be replaced by an arbitrary proper morphism but $S^\cdot$ must correspond to a vector bundle).

He states (1) not for projective cones, but for cones (replace $\textrm{Proj}$ above with $\textrm{Spec}$) that moreover have the natural map $\mathcal{O}_X \rightarrow S^0$ an isomorphism (even though he gives the more general definition above). To work out the details of something in his book I'd like this stronger result above. I'm asking for a reference in case such a result is actually false in this stronger setting.

$$---$$ Just to be concrete, here is what I'm trying to work out.

(Fulton Intersection Theory, Example 4.1.6b) Let $X$ be a scheme of finite type over a field, and $C$ be a cone on $X\times \mathbb{A}^1$ flat over $\mathbb{A}^1$. Let $i_t: X \rightarrow X \times \mathbb{A}^1$ be the inclusion of $X$ into the product at $t$. Then, $i_t^*s(C) = s(C_t)$. (Note that flattness would be automatic if the cone had support equal all of $X \times \mathbb{A}^1$, so Fulton isn't assuming full support.) Here, $C_t$ denotes the restriction of the cone to $X_t = X \times \{t\}$.

Pf:

Let $i_{C_t}: P(C_t \oplus 1) \rightarrow P(C \oplus 1)$ be the closed immersion predicted by (1).

After standard manipulations, we need to show that (for fixed integer $n$):

$$i_{C_t}^*\left( c_1( \mathcal{O}(1))^n \cap [P(C\oplus 1)]) \right) = c_1( \mathcal{O}(1))^n \cap [P(C_t\oplus 1)]$$

where the LHS and RHS $\mathcal{O}(1)$ denote the corresponding sheaves on $P(C\oplus 1)$ and $P(C_t \oplus 1)$ respectively.

Now, if we had (2) above, this would be true by Prop 2.6(e) (How gysin map acts on Chern classes).