  We theoretically study topological planar Josephson junctions (JJs) formed
from spin-orbit-coupled two-dimensional electron gases (2DEGs) proximitized by
two superconductors and subjected to an in-plane magnetic field $B_\parallel$.
Compared to previous studies of topological superconductivity in these
junctions, here we consider the case where the superconducting leads are
narrower than the superconducting coherence length. In this limit the system
may be viewed as a proximitized multiband wire, with an additional knob being
the phase difference $\phi$ between the superconducting leads. A combination of
mirror and time-reversal symmetry may put the system into the class BDI.
Breaking this symmetry changes the symmetry class to class D. The class D phase
diagram depends strongly on $B_{\parallel}$ and chemical potential, with a
weaker dependence on $\phi$ for JJs with narrower superconducting leads. In
contrast, the BDI phase diagram depends strongly on both $B_\parallel$ and
$\phi$. Interestingly, the BDI phase diagram has a "fan"-shaped region with
phase boundaries which move away from $\phi = \pi$ linearly with $B_\parallel$.
The number of distinct phases in the fan increases with increasing chemical
potential. We study the dependence of the JJ's critical current on
$B_\parallel$, and find that minima in the critical current indicate
first-order phase transitions in the junction only when the spin-orbit coupling
strength is small. In contrast to the case of a JJ with wide leads, in the
narrow case these transitions are not accompanied by a change in the JJ's
topological index. Our results, calculated using realistic experimental
parameters, provide guidelines for present and future searches for topological
superconductivity in JJs with narrow leads, and are particularly relevant to
recent experiments [A. Fornieri et al., Nature (London) 569, 89 (2019)].
