Basic questions on the Hilbert scheme/ Douady space Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More precisely, it represents the functor $F$ from schemes (resp. complex analytic spaces) to sets given by 
$$F(S)=\{Y\subset X\times S|\, Y \mbox{- closed subscheme (resp. anal. subspace), flat proper over } S\}.$$
Question 1. Let $X_1\subset X$ be a closed subscheme. Is it true that the Hilbert scheme of $X_1$ is a closed subscheme of the Hilbert scheme of $X$? The same question for the Douady spaces.
Question 2. I would like to have a modification of the Hilbert scheme/ Douady space which parameterizes closed subschemes of $X$ with a marked point. I guess one should consider the functor 
$$G(S)=\{Y\subset X\times S \mbox{ as above and a morphism } f\colon S\to Y|\, p\circ f=Id_S\},$$ 
where $p\colon Y\to S$ is the natural projection. Is the functor $G$ representable?
Question 3. Assume that Questoin 2 has a positive answer, namely the functor $G$ is representable by a scheme (resp. complex analytic space) $\mathcal{F}$. Obviously we have the morphisms $\mathcal{F}\to X$ and $\mathcal{F}\to Hilb(X)$ which forget either the subscheme or the marked point. Is it true that the induced morphism $\mathcal{F}\to X\times Hilb(X)$ is a closed imbedding? The same question for Douady space.
I believe answers to all these questions are well known to experts. Is there a standard reference to this material? The complex analytic case is more important for me, but the algebraic case is also of interest.
 A: Let me give an answer for Hilbert schemes.
Q. 1. Yes, the reason is that the functor $F_{X_1}$ is a closed subfunctor of the functor $F_X$ with respect to the natural embedding. In other words, for each $S$ and each $Y \in F_X(S)$ the maximal subscheme $S_1 \subset S$ such that $Y\times_S S_1 \in F_{X_1}(S_1)$ is closed in $S$. In fact subscheme $S_1$ can be defined as the zero locus of the canonical morphism $\pi_*I_{S\times X_1}(n) \to \pi_*O_Y(n)$ obtained by pushing forward the composition $I_{S\times X_1} \subset O_{S\times X_1} \to O_Y$ twisted by the sheaf $O_X(n)$ with $n$ sufficiently large (so that $I_{X_1}(n)$ is globally generated) via the projection $\pi:S\times X \to S$. The reference here is Grothendieck's FGA (or more recent "FGA explained").
Q. 2. I think it is more natural to consider so-called flag Hilbert scheme which parameterizes flags of subschemes $Y' \subset Y \subset S\times X$. If you specify the Hilbert polynomial of (fibers of) $Y'$ to be constant 1, this will give you precisely what you want. Representability then is known. I do not know a reference, but I think you can find one by googling for "flag Hilbert scheme".
Q. 3. Yes, this is a general fact for flag Hilbert schemes. Alternatively you can argue in the same way as in Q. 1.
I am completely sure that the same arguments should work for Douady spaces.
A: To your second and third question:
Let $Z \subset Hilb(X) \times X$ be the universal family over $Hilb(X)$. Then, immediately, $Z$ represents the functor you are asking for.
