  The $\phi$-Assouad dimensions are a family of dimensions which interpolate
between the upper box and Assouad dimensions. They are a generalization of the
well-studied Assouad spectrum with a more general form of scale sensitivity
that is often closely related to "phase-transition" phenomena in sets.
  In this article we establish a number of key properties of the $\phi$-Assouad
dimensions which help to clarify their behaviour. We prove for any bounded
doubling metric space $F$ and $\alpha\in\mathbb{R}$ satisfying
$\overline{\operatorname{dim}}_{\mathrm{B}}F<\alpha\leq\operatorname{dim}_{\mathrm{A}}
F$ that there is a function $\phi$ so that the $\phi$-Assouad dimension of $F$
is equal to $\alpha$. We further show that the "upper" variant of the dimension
is fully determined by the $\phi$-Assouad dimension, and that homogeneous Moran
sets are in a certain sense generic for these dimensions.
  Further, we study explicit examples of sets where the Assouad spectrum does
not reach the Assouad dimension. We prove a precise formula for the
$\phi$-Assouad dimensions for Galton--Watson trees that correspond to a general
class of stochastically self-similar sets, including Mandelbrot percolation.
This result follows from two results which may be of general interest: a sharp
large deviations theorem for Galton--Watson processes with bounded offspring
distribution, and a Borel--Cantelli-type lemma for infinite structures in
random trees. Finally, we obtain results on the $\phi$-Assouad dimensions of
overlapping self-similar sets and decreasing sequences with decreasing gaps.
