We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE$(α,η)$ of continuum planar aggregation models. The class includes regularized versions of the Hastings--Levitov family HL$(α)$ and continuum versions of the family of dielectric breakdown models, where the local attachment intensity for new particles is specified as a negative power $-η$ of the density of arc length with respect to harmonic measure. The limit dynamics follow solutions of a certain Loewner--Kufarev equation, where the driving measure is made to depend on the solution and on the parameter $ζ=α+η$. Our results are subject to a subcriticality condition $ζ\le1$: this includes HL$(α)$ for $α\le1$ and also the case $α=2,η=-1$ corresponding to a continuum Eden model. Hastings and Levitov predicted a change in behaviour for HL$(α)$ at $α=1$, consistent with our results. In the regularized regime considered, the fluctuations around the scaling limit are shown to be Gaussian, with independent Ornstein--Uhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if $ζ\le1$.