  We study the prime-to-$p$ Hecke action on the projective limit of the sets of
connected components of Shimura varieties with fixed parahoric or Bruhat--Tits
level at $p$. In particular, we construct infinitely many Shimura varieties for
CM unitary groups in odd variables for which the considering actions are not
transitive. We prove this result by giving negative examples on the question of
Bruhat--Colliot-Th\'el\`ene--Sansuc--Tits or its variant, which is related to
the weak approximation on tori over $\mathbb{Q}$.
