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Models of composite Higgs provide a very compelling solution to the hierarchy problem and are among the most popular extensions of the Standard Model. However, current bounds on top partners have put them under some pressure, hinting to a scale of new physics that may be beyond the reach of the LHC. In this talk, I will discuss two different scenarios where composite Higgs models could be first probed in non-resonant searches. In the first one, the strong dynamics can manifest first in the observation of violation of lepton flavor universality, as suggested by the recent anomalies on $R_K$ and $R_K^{\ast}$, offering a beautiful interplay between the latter and naturalness. In the second one, I will discuss a non-minimal model featuring a singlet or an electroweak triplet WIMP, where direct and indirect dark matter searches prove to be an excellent complement to collider searches.