  Cosmological models beyond $\Lambda$CDM, like those featuring massive
neutrinos or modifications of gravity, often display a characteristic change
(scale-dependent suppression or enhancement) in the matter power spectrum when
compared to a $\Lambda$CDM baseline. It is therefore a widely held view that
constraints on those models can be obtained by searching for such features in
the clustering statistics of large-scale structure. However, when using biased
tracers of matter in the analysis, the situation is complicated by the fact
that the bias also depends on cosmology. Here we investigate how the selection
of tracers affects the observed signatures for two examples of
beyond-$\Lambda$CDM cosmologies: massive neutrinos and clustering dark energy
($k$-essence). We study the signatures in the monopole, quadrupole, and
hexadecapole of the redshift-space power spectra for halo catalogues from large
$N$-body simulations and argue that a fixed selection criterion based on local
attributes like tracer mass leads to a near loss of signal in most cases.
Instead, the full signal is recovered only if the selection of tracers is done
at fixed bias. This emphasises the need to model or measure the bias parameters
accurately in order to get meaningful constraints on the cosmological model.
