The wonderful responses to an earlier question Self-intersection and the normal bundle motivated me to ask the following question:

Let $Y \subset X$ be a subvariety of a variety $X$. Infinitesimal deformations of $Y$ in $X$ are subschemes of $X \times \textrm{Spec } k[\epsilon]/(\epsilon^2)$ flat over $\textrm{Spec } k[\epsilon]/(\epsilon^2)$ and with closed fiber $Y$. Such subschemes correspond bijectively to sections of the normal bundle $\mathcal{N}_{Y/X}$. (Hartshorne, III.9)

$\textbf{Question:}$ Do infinitesmal deformations of a regularly embedded subvariety $Y \subset X$ of codimension $d$ naturally determine cycles in $X$ (rationally equivalent to $Y$)? This seems like a bit of a long shot, but comments of Charles and Donu in the linked question seem to suggest that something like this is true.

If this were true, it would be important for both the linked question, and in it's own right. References where I can learn the relevant material be greatly appreciated.