  We give a complete description of partially wrapped Fukaya categories of
graded orbifold surfaces with stops. We show that a construction via global
sections of a natural cosheaf of A$_\infty$ categories on a Lagrangian core of
the surface is equivalent to a global construction via the (equivariant) orbit
category of a smooth cover. We therefore establish the local-to-global
properties of partially wrapped Fukaya categories of orbifold surfaces closely
paralleling a proposal by Kontsevich for Fukaya categories of smooth Weinstein
manifolds.
  From the viewpoint of Weinstein sectorial descent in the sense of Ganatra,
Pardon and Shende, our results show that orbifold surfaces also have Weinstein
sectors of type $\mathrm D$ besides the type $\mathrm A$ or type
$\widetilde{\mathrm A}$ sectors on smooth surfaces.
  We describe the global sections of the cosheaf explicitly for any generator
given by an admissible dissection of the orbifold surface and we give a full
classification of the formal generators which arise in this way. This shows in
particular that the partially wrapped Fukaya category of an orbifold surface
can always be described as the perfect derived category of a graded associative
algebra. We conjecture that associative algebras obtained from dissections of
orbifold surfaces form a new class of associative algebras closed under derived
equivalence.
