You need to know about this before you work in any electrotechnical area, this is one of the most important and basic aspects of circuits. There, patronising over, sorry but it's true and needed saying if you are undertaking electrical work.
The essential aspect of 'fusing' (fuses, MCBs etc) ADS, where utilised is that all conductors must be protected for their intended/designed use in relation to location and loading. Use the bs7671(2018) to refer to the required cable selection if you are unsure.
Provided the 16amp circuit is protecting 2.5mm conductors or greater surface clipped in twin and earth etc there is a 'good chance' that those conductors will be protected by it (depends on run length), however, if smaller cross sectional area conductors are utilised, or the route means the de-rating is too great, then you may need to change the breaker or add an fcu at some point (as was there originally) to suitably protect the non protected conductors.
This applies regardless of any other circumstances, so you will need to assess both the rigs manufacturer reccomended protection, and the actual maximum load characteristics of the 'rigs' wiring, to be able to safely install the new 'plug', to the 'rig',
In reality it's two assessments, 1) suitability for the socket on the 16 amp circuit. 2) suitability for the lighting rig to be connected by an unfused plug.
If either is unsuitable, then the conductors will need suitable protection adding/changing (fcu, mcb etc) or conductors upgrading to larger cross sectional area.
All the best.
