Suppose X is a smooth projective variety, say over $\mathbb{Q}$ for simplicity. Let $F$ be a finite extension of $\mathbb{Q}$. Let $\mathrm {Ch}^{r}(X/F)$ denote the Chow group of codimension $r$ algebraic cycles defined over $F$. A conjecture of Tate asserts that the cycle class map from $\mathrm{Ch}^r(X/F)$ to $H^{2r}_{et}(X)(r)$ is injective, with image in the subspace fixed by $\mathrm{Gal}(\overline{\mathbb{Q}}/F)$. In particular, the dimension of $\mathrm{Ch}^r(X/F)$ should be uniformly bounded (by the $2r^{\mathrm{th}}$ Betti number of $X(\mathbb{C})$) as $F$ varies.

On the other hand, let $\mathrm{Ch}^{r}(X/F)_0$ denote the kernel of the cycle class map, which is to say the group of homologically trivial cycles of codimension $r$ modulo rational equivalence. The dimension of this guy is predicted by Beilinson and Bloch to be given as the order of vanishing of $L(s,H^{2r-1}(X/F))$ at its central critical point. Now, the order of vanishing of this L-function can by made to increase very rapidly as $F$ varies; for example, one could (in some circumstances) choose $F$ to be an abelian extension such that each of the twists $L(s,H^{2r-1}(X/\mathbb{Q})\times \chi)$ has root number $-1$ for $\chi$ varying over characters of $\mathrm{Gal}(F/\mathbb{Q})$. When $X$ is an elliptic curve and $r$=1, this phenomenon has been confirmed in a variety of situations: for $\mathbb{Z}/p^{n}\mathbb{Z}$-towers over imaginary quadratic fields (Cornut-Vatsal), for Hilbert class fields of imaginary quadratic fields (Templier), and for towers of Kummer extensions (Darmon-Tian). However, for higher dimensional varieties and higher codimension cycles, the relevant L-functions aren't even well understood.

My question: is there a "conceptual" reason why there should have lots of homologically trivial cycle classes over extensions of the base field? In other words, if you believe certain conjectures about L-functions, then this is not hard to guess, but I am looking for some motivation which avoids L-functions.

(Edited in response to a comment of moonface.)