Can I move a running application to a different X server?

If you look at the Wikipedia page on the subject there are several apps mentioned.

Xmove

excerpt

xmove is a computer program that allows the movement of X Window
System applications between different displays and the persistence of
X applications across X server restarts.[4] It solves a problem in the
design of X, where an X client (an X application) is tied to the X
server (X display) it was started on for its lifetime. Also, if the X
server is shut down, the client application is forced to stop running.

xmove lets the client disconnect from its current X server, and
connect to a new one, at any time. The transition is completely
transparent to the client. xmove works by acting as a proxy between
the client and server. It is a “pseudoserver” which stores enough
server state so that clients can connect to a new server without being
disrupted.

Xpra

excerpt

xpra or X Persistent Remote Applications is a tool which allows you to
run X clients usually on a remote host and then direct their display
to your local machine without losing any state.1

It differs from standard X forwarding in that it allows disconnection
and reconnection without disrupting the forwarded application. It
differs from VNC and similar remote display technologies in that xpra
is rootless: i.e., applications forwarded by xpra appear on your
desktop as normal windows managed by your window manager, rather than
being all “trapped in a box together”. Xpra also uses a custom
protocol that is self-tuning and relatively latency-insensitive, and
thus is usable over worse links than standard X.

Guievict

excerpt

guievict is a computer program which enables the GUI of any
application for XFree86 implementation of X Window to be transparently
migrated to or replicated on another display. Unlike some program
providing similar functionalities, it requires neither prearranging
steps such as re-linking the application program binary nor
re-directing the application process’s window system communication
through a proxy like xmove does.

Guievict is based on a small X server extension that enables an
application to retrieve its window state from the X server and a
library of GUI migration functionality that is injected in the
application process at run time. Code injection or runtime
code-patching can be done via the DynInst API. However, guievict
contains its own implementation to avoid requiring users to install
DynInst.

Of the 3 of these, Guievict sounds like what you’re looking for, mainly that it can checkpoint the state of X application AppX and migrate it to another X server where it can be restored.