Let us consider the following situation: $S$ is smooth and quasiprojective variety (over $\mathbb{C}$) and $p\colon X\to S$ is a smooth and proper morphism. I denote by $X_t$ the fibre over $t\in S$.
Suppose that we have got a $p$-cycle $Z_t$ on $X_t$, for every $t\in S$.
What are the obstructions to gluing the $Z_t$s to a cycle $Z$ on $X$?
This is pretty vague, I try to expand a bit more.
In literature I have often found the statement that the $Z_t$ can always be glued to form a cycle with rational coefficients. This is justified by the fact that the gluing process is done after a finite change of base.
The intuition I built over these statement is that we could have monodromy phenomena. More precisely, I can start in a point $t$ and start moving around $S$ - if eventually I come back to $t$ (I followed a loop), and the monodromy in $p$ is non-trivial, I may end up with a "different" cycle (i.e. the glued cycle $Z$ would have an indeterminacy at $t$). Now, if I know that the family has finite monodromy, a finite base change would surely solve this indeterminacy.
But this is not a rigorous argument and moreover, it involves the knowledge of the monodromy of the family, while the result seems to be true in more generality.
So I end up with two questions:
1) Is my intuition right?
2) What is a rigorous argument to prove the statement that the $Z_t$s can be glued to form a cycle $Z$ on $X$, with rational coefficients? (References are very welcome. Otherwise, also sketch of proofs are very welcome - I am happy to fill the gaps (it'd be a good exercise indeed).)
Also, if my intuition is true, it looks to me that the glued cycle $Z$ should have integral coefficients as soon as the base $S$ is simply-connected. In particular, we can always find a (infinitesimal) open subset of $S$ where $Z$ has integral coefficients. Is it right?
P.S.: I haven't said in which category I am working. It can be either the category of analytic spaces or algebraic varieties (in the last case I guess one should talk about $\operatorname{Spec} DVR$ instead of infinitesimal open subset).
I have never used the word Chow group, because it seems to me that I do not need to work up to rational equivalence. Nevertheless, it may be the case that I am misunderstanding something...
EDIT 1
Let me put a concrete example. In the proof of Lemma 1.2 of this paper, the authors construct cycles $\mathcal{Z}'_t$ (using their notations) on each fibre. Then they say that a "standard argument shows that for an adequate choice of $N$ they can be constructed in families [...] by spreading the original cycles".
This is the motivation for my question. If you explain to me what is this "standard argument", it would probably answer my question.
EDIT 2
Following Sam's comment, I looked at Voisin's book (and other sources: e.g. Green and Griffiths-Green). It looks like what I had written, about monodromy and the hope to glue the cycles infinitesimally, is wrong. Before declaring the question closed though, I would appreciate if anyone can add something more about the argument. It looks like a classic and important step to understand and I would like to earn as much as possible from this post.
Thank you!