@AJMB It looks like you're asking if the effects applied to individual clips in the original sequence carry over into the multi-cam sequence you created from it. Yes, they do. If effects do not appear in the multicam sequence, go back to the original sequence and verify that all the tracks are on (eyeball) and the effects are turned on in the clips' 'Effects control panel.'
You said "putting together a multicam sequence." A multicam is a type of 'nested' sequence made from a multitrack sequence. First , you should have a sequence with 2 or more tracks sync'ed together (including audio). You can apply effects to these track clips now, but you should wait 'til after you've rough edited the multicam.
You make a multicam from that original sequence by dragging its icon to the 'new item' icon (or Rt-clicking and "New sequence from clip") The new sequence will have just one green clip. Rt click that clip and click 'Enable' from the 'Multi Camera' dropdown. This is what activates the multicam sequence.
In the Window dropdown , check 'Multi Camera monitor' and do your 'live switching.' When you're done Premiere places cuts at each edit point where you switched cameras. Each segment represents the appropriate track in the original sequence, including any effects applied.
You can also apply effects to the individual segments in the multicam track, or an 'adjustment layer', but it could get confusing if you also have effects on the originals.
This is probably more info than you needed, but other people read these. Let me know if still having pproblems