A semi-empirical model is presented that describes the development of a fully developed turbulent boundary layer in the presence of surface roughness with length scale $k_{s}$ that varies with streamwise distance $x$. Interest is centred on flows for which all terms of the von Kármán integral relation, including the ratio of outer velocity to friction velocity $U_{\infty }^{+}\equiv U_{\infty }/u_{\unicode[STIX]{x1D70F}}$, are streamwise constant. For $Re_{x}$ assumed large, use is made of a simple log-wake model of the local turbulent mean-velocity profile that contains a standard mean-velocity correction for the asymptotic fully rough regime and with assumed constant parameter values. It is then shown that, for a general power-law external velocity variation $U_{\infty }\sim x^{m}$, all measures of the boundary-layer thickness must be proportional to $x$ and that the surface sand-grain roughness scale variation must be the linear form $k_{s}(x)=\unicode[STIX]{x1D6FC}x$, where $x$ is the distance from the boundary layer of zero thickness and $\unicode[STIX]{x1D6FC}$ is a dimensionless constant. This is shown to give a two-parameter $(m,\unicode[STIX]{x1D6FC})$ family of solutions, for which $U_{\infty }^{+}$ (or equivalently $C_{f}$) and boundary-layer thicknesses can be simply calculated. These correspond to perfectly self-similar boundary-layer growth in the streamwise direction with similarity variable $z/(\unicode[STIX]{x1D6FC}x)$, where $z$ is the wall-normal coordinate. Results from this model over a range of $\unicode[STIX]{x1D6FC}$ are discussed for several cases, including the zero-pressure-gradient ($m=0$) and sink-flow ($m=-1$) boundary layers. Trends observed in the model are supported by wall-modelled large-eddy simulation of the zero-pressure-gradient case for $Re_{x}$ in the range $10^{8}{-}10^{10}$ and for four values of $\unicode[STIX]{x1D6FC}$. Linear streamwise growth of the displacement, momentum and nominal boundary-layer thicknesses is confirmed, while, for each $\unicode[STIX]{x1D6FC}$, the mean-velocity profiles and streamwise turbulent variances are found to collapse reasonably well onto $z/(\unicode[STIX]{x1D6FC}x)$. For given $\unicode[STIX]{x1D6FC}$, calculations of $U_{\infty }^{+}$ obtained from large-eddy simulations are streamwise constant and independent of $Re_{x}$ when this is large. The present results suggest that, in the sense that $U_{\infty }^{+}(\unicode[STIX]{x1D6FC},m)$ is constant, these flows can be interpreted as the fully rough limit for boundary layers in the presence of small-scale linear roughness.