Macbook pro M1 and Imac 27 inch (IOS Mojave) as a second screen

Unfortunately, that setup is not supported by Apple. The iMac cannot be used as a monitor for any of the newest Macs and Target Display mode is no longer a feature of macOS, having been deprecated several years ago.

Please see: Use your iMac as a display with target display mode - Apple Support

Per the link above:

• Your target iMac (monitor) must be using macOS High Sierra 10.13.6 or earlier. 

• In all cases the source Mac that you're connecting FROM must have been introduced in 2019 or earlier and have macOS 10.15 Catalina or earlier installed.

However, you might consider a third-party solution, for example: Luna Display or Duet Display, though I can’t speak for the performance of those.

Do yourself a favor, avoid the headache and just budget an extra $100 for a monitor. Yes, they are that inexpensive.

Your M1 MacBook Pro doesn't support using ANY iMac as a TDM display, regardless of whether that iMac satisfies the requirements for TDM.

Your 27" Late 2013 iMac could act as a TDM display IF it were running High Sierra or earlier, and IF you connected it to a much older Mac than that M1 MacBook Pro. But that doesn't help you here.

Use your iMac as a display with target display mode - Apple Support

Like D.I. Johnson, I have not used Luna Display but I recently noted this performance-related advisory on the Luna Displays home page:

To me that says they expect lagginess.