Omega has something call \localleftbox and \localrightbox which are attached to every line of a paragraph but which can only hold a fixed content.

What is often wanted is to compute this content on a per line basis which would require to execute macro code while the broken paragraph is being moved onto the current vertical list.

A number of ways to achieve this:

in §880 before actually moving the line to the current vertical list, start a new hlist and expand/execute some token register \everyleftline as it contents. Once that box is assembled package it and overlay its reference point with the reference point of the line and attach both on the vertical list (i.e., the \hbox will not take up any space)

rather than actually putting the assembled in onto the current vertical list, put it into a special box register (say \currlinebox) and simply execute a token register called \everyline which has the default definition \box\currlinebox.

open question: what to do with material that has been moved out of the box (eg from \vadjust) .. several options, one could be to have those special notes put into a \vbox, say, \currlineaftermaterial and then make the default definition

of course all this needs some checking that \currlinebox (which internally is call "justbox" in the program) is not accessed outside \everyline ... and then there is the big question by Thanh: can we execute user supplied macro material in such places?