According to your comments, it seems to be driver issue. I suggest you to install the latest NUC drivers according to your operating system and latest driver provided with Logitech.
I recommend you to install the latest BIOS on each NUC.
Latest BIOS on 5i5ryk is version 0352 and D54250wyk is v. 0041
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This does not seem to be a driver issue since the mouse and keyboards plugged directlyinto the NUC's 3.0 usb key(s) work properly.
It is recommended to use straight connections with high quality cables to ensure best performance. The use of kvm switches, usb display adapters, and VGA/DVI/HDMI Adapters is possible, however, we could not guarantee its functionality on all the cases. The same happens to Logitech Unifying dongle* it depends on it specifications and the maximum allowed paired devices. It would be good to contact Logitec* and confirm if your mouse and keyboard are compatible and if they have any suggested workaround for this issue. unifying dongle - Logitech Forums
Intel® NUC Board NUC5i5RYB's and Intel® NUC Board NUC5i3RYB's boards support eight USB ports. All eight ports are high-speed, full-speed, and low-speed capable. The port arrangement is as follows:
• USB 3.0 ports:
+Two ports are implemented with external front panel connectors (blue and orange).
+Two ports are implemented with external back panel connectors (blue).
• USB 2.0 ports:
-Two ports via one dual-port internal 1x8 1.25 mm pitch header (white).
-One port is reserved for an M.2 1216 for the Wireless module.
Please take into consideration the board has a power sensing circuit that:
• Manages CPU power usage to maintain system power consumption below 65 W.
• Is designed for use with 65 W AC-DC adapters.
You could check the PSU(Power Supply) Wattage and confirm if it is more than 65W then it is recommended that you disable this feature (via BIOS option) when using an AC-DC adapter greater than 65 W which may prevent the device from providing full power to USB ports .
You could also test your Logitech Unifying dongle.*
You may get more information here https://downloadmirror.intel.com/24565/eng/NUC5i5RYB_NUC5i3RYB_TechProdSpec02.pdf
*Other names and brands may be claimed as the property of others.