  We consider the Cauchy problem \begin{align*} \partial_t u+u\partial_x
u+L(\partial_x u) &=0, \\ u(0,x)=u_0(x) \end{align*} on the torus and on the
real line for a class of Fourier multiplier operators $L$, and prove that the
solution map $u_0\mapsto u(t)$ is not uniformly continuous in $H^s(\mathbb{T})$
or $H^s(\mathbb{R})$ for $s>\frac{3}{2}$. Under certain assumptions, the result
also hold for $s>0$. The class of equations considered includes in particular
the Whitham equation and fractional Korteweg-de Vries equations and we show
that, in general, the flow map cannot be uniformly continuous if the dispersion
of $L$ is weaker than that of the KdV operator. The result is proved by
constructing two sequences of solutions converging to the same limit at the
initial time, while the distance at a later time is bounded below by a positive
constant.
