The problem is that Pegasus Mail updates configuration files each time it is opened and closed. Those configuration files reside inside of the mailbox directory. Because of this, the mailbox directory can not be accessed by two instances of Pegasus Mail at the same time. This is key: two instance of Pegasus Mail can not access the same mailbox at the same time. The result often is corruption of one or more files in the mailbox directory.
If the mailboxes are on one machine then the devices must always have access to this machine in order to do email. There is a concept of New mail mailbox and Home mailbox whereby new mail is stored locally until moved to a folder at which time it is moved to the home mailbox. This might work for you but requires that users keep their new mail folder empty in order for all mail to be accessible by other devices. The limitation of two instances simultaneously accessing a mailbox remains.
Finally, Pegasus Mail is multi-user capability meaning that a single installation can be run by multiple users. In my office I have a single instance of Pegasus Mail installed on a server with 13 users running that instance on their desktop PCs. Each user can run their configuration from whatever office PC they need to but the caveat remains that they can run only one instance at a time. The problem with this setup is that laptops, tablets, and smartphones need access to the mail when not connected to the office LAN (that's where running an IMAP capable mail server comes in). If all of your family devices will only access mail when connected to the home LAN then the multi-user approach would work except for devices that can not run an app like Pegasus Mail (eg; smartphones and iDevices).