Hi Mike,
A PixInsight project consists of a .xosm file and a .data folder. For example, if you save a "MyBestImageEver" project, you'll get the following two items on the folder where you saved the project:
MyBestImageEver.xosm
MyBestImageEver.data
The .data folder is where all the project data are stored: images, icons, processing histories, tool settings, text files, etc. The .xosm file describes your project, all the associated data and their relations.
XOSM stands for XML Object Serialization Module, and is PixInsight's project format. The XOSM format is fully relocatable. This means that you can move or copy a project to any location on your filesystem, or move it to another machine, and it will continue working without problems.
This is true for internal project data, such as open images. However, if your project contains 'hardwired' absolute file paths---for example, ImageContainer icons, or file lists in tools such as ImageIntegration, StarAlignment, BatchPreprocessing, etc---, PixInsight has no way to know where these files are (in case they still exist) if you copy or move your project to a different machine, so the items that depend on such files won't work and fail with 'no such file' messages.
There are several ways to prevent and/or fix these problems. The best way is keeping all of your PixInsight projects and data on external storage devices, which you can mount at the same point on different machines. For example, if you use Windows, you can assign the same drive letter to an external disk on two or more PCs. On UNIX and Linux, removable devices can be mounted at arbitrary locations on the filesystem, which allows for more flexible configurations.
If the above is not possible for some reason, you can always edit the XOSM file to replace nonexistent absolute file paths with valid paths. Since XOSM is plain text, it can be edited with any good code editor, such as PixInsight's Script Editor.