  The heavy-fermion metal YbRh$_2$Si$_2$ realizes a field-induced quantum
critical point with multiple vanishing energy scales $T_{\rm N}(B)$ and
$T^\ast(B)$. We investigate their change with partial non-isoelectronic
substitutions, chemical and hydrostatic pressure. Low-temperature electrical
resistivity, specific heat and magnetic susceptibility of
Yb(Rh$_{1-x}$T$_x$)$_2$Si$_2$ with T=Fe or Ni for $x\leq 0.1$, magnetic fields
$B\leq 0.3$~T (applied perpendicular to the c-axis) and hydrostatic pressure
$p\leq 1.5$~GPa are reported. The data allow to disentangle the combined
influences of hydrostatic and chemical pressure, as well as non-isoelectronic
substitution. In contrast to Ni- and Co-substitution, which enhance magnetic
order, Fe-substitution acts oppositely. For $x=0.1$ it also completely
suppresses the $T^\ast$ crossover and eliminates ferromagnetic fluctuations.
The pressure, magnetic field and temperature dependences of $T^\ast$ are
incompatible with its interpretation as Kondo breakdown signature.
