I have a Lie group $G$, an orbit manifold $Y = G/H$ for some closed subgroup $H$, and a $G$-equivariant map $p : X \to Y$, where $X$ is a manifold with a smooth $G$-action (not assumed to be transitive or free) with closed orbits. Is the map $p$ automatically a fiber bundle?
My intuition is that the $G$-action makes not only the space $Y$ but also the map $p$ homogeneous, so a cover trivializing the fiber bundle $G \to Y$ should also trivialize $p$. But actually implementing this requires "uniformizing" the action of $H$ on $X$ restricted to such trivializing neighborhoods, which seems not obvious especially when there are stabilizers. As a sanity check, we get that the fibers of $p$ are diffeomorphic: $gp^{-1}(y) = p^{-1}(gy)$. When the action of $G$ on $X$ is also transitive, $p$ is a fiber bundle, as explained in Criterion for equivariant maps to be fiber bundles?.
I am happy to further assume that the action on $X$ has finite point-stabilizers $\operatorname{Stab}_G(x) = \operatorname{Stab}_H(x)$ and that these stabilizers are trivial for $x$ generic (ie in some dense open set), but I don't see how this would be useful. $G$ need not be compact, but a proof assuming $Y$ to be compact is acceptable. If $X$ is compact (or more generally if $p$ is known to be proper) then we can conclude that $p$ is a fiber bundle from Ehresmann's theorem since the assumptions force $p$ to be a submersion: $(g \mapsto g \cdot p(x)) = p \circ (g \mapsto g \cdot x)$ and the left side is a submersion by choice of $Y$, in particular the derivative at $g = e$ is surjective.
I am particularly interested in the case of smooth complex varieties and complex algebraic groups (with algebraic actions). A class of examples is given by taking $G = \mathrm{GL}(\mathbf{C}, n)$, $Y$ the Grassmannian $\mathrm{Gr}_{\mathbf{C}}(n, r)$ and $X$ the incidence variety of smooth degree $d$ homogeneous polynomials and $r$-planes $$\{(f, \gamma) \in \mathbf{C}[x_1, \dots, x_n]_d \times Y \mid f|\gamma = 0, V(\partial_i f) = \{0\}\}$$ for some $d \ge 3$ (so that the stabilizers are finite) but not too large (so that $X$ is non-empty).
$p$ is a fiber bundle without assuming the $G$ action on $X$ has closed orbits. In fact we can write down a local trivialization of $p$ in terms of a local trivialization of $G \to G/H$.
Consider a local trivialization of $G \to Y$ over a neighborhood $U$ in $Y$, ie a diffeomorphism $\varphi : U \times H \to G|_U = \{g \in G \mid gH \in U\}$ such that $\varphi(y, h) H = y$. Let $\psi : U \to G$ be defined by $\psi(y) = \varphi(y, e)$ (here $e$ could be replaced by any fixed $h_0 \in H$). Let $y_0 = eH \in G/H = Y$. Therefore $\psi(y) y_0 = y$ for each $y \in U$.
Now $U \times p^{-1}(y_0) \cong p^{-1}(U)$ via the inverse pair of diffeomorphisms $(y, x_0) \mapsto \psi(y) x_0$ and $x \mapsto (p(x), \psi(p(x))^{-1} x)$, which is a local trivialization for $p$.