If two stars form from the giant molecular cloud it is usually assumed that their chemical abundances, including the chemical abundances in the photosphere - which is what is measured by stellar spectroscopy, will be the same. This is because it is thought that mixing and turbulence within the cloud are efficient enough to chemically homogenise it.
Of course we can try to measure small differences in chemical abundances, via the spectroscopy of stars. There have been studies of stars within clusters that have shown that if there is any chemical inhomogeneity, it is very small (e.g. Wilden et al. 2002). These studies are normally carried out in the context of trying to see whether the stars have accreted planets or other material - i.e. the assumption is that the stars have a uniform chemistry, so that any variations seen would actually reflect differences in what they have "eaten".
Some chemical elements have a notable dispersion among stars born from the same cloud. Lithium is a notable example, which is burned to helium at relatively low temperatures and hence depleted from the star as they age. i.e. The dispersion is because the Li has been processed at different rates within the stars, not because these stars started off with differing Li abundances.
If one were looking for small variations in abundances it isn't clear to me why you would think to look for those variations in "trace elements". These are the elements with the smallest chemical abundances and usually among the most difficult to measure with any precision. The differences would be much more obvious in the iron, nickel, silicon or magnesium abundances, which have plenty of measurable spectral absorption lines in stars like the Sun.
Finally, I'm not sure what you mean by a "runaway effect" on the composition. The chemical abundances in a star's atmosphere when it is born are largely the same chemical abundances seen when it reaches the end of the main sequence. In a star like the Sun, there are no mixing processes which are able to change the observable photospheric abundances very much (apart from lithium) during main sequence evolution. This is because all that happens in a main sequence star is that hydrogen is turned into helium. Most of that extra helium sinks into the core and does not "dilute" the surface abundances. In any case, helium is not an element with easily recognisable spectroscopic signatures in the atmosphere of a star like the Sun. This may not be the case in more massive stars with convective cores which burn hydrogen through the CNO cycle. But here, subsequent differences in abundances are more likely telling us about differing masses of the stars or how differing rotation rates affect mixing.
Thus even if star were born with slightly different compositions, these differences would remain the same throughout most of their lives. Larger composition differences would affect the appearances of the stars (through the effect on atmospheric opacities) and result in stars of the same mass adopting different positions in the Hertzsprung-Russell diagram at the same age. This has been most notably observed in globular clusters, where there can be multiple populations of stars with different chemical compositions (e.g. Gratton et al. 2012), although it is unclear whether these differences are intrinsic to the cloud or arise from pollution of some forming stars with the ejecta from others.