Sounds like this could be several different issues at play here. First, I’ve had problems with PNG files in FCP, so I avoid them. I use TIFFs or hires JPGs. In a full 1080 sequence JPGs are usually fine as long as they are hi res. Also the frame dims of the image sequence should be optimal. This means your stills (PNGs) should be close or exactly your sequence dims. FCP will handle them better.
I’m also not so sure about your workflow. Why are you transcoding to an Image Sequence with Quicktime? A single video file will be more efficient. I don’t know Maya, but can you output video clips that are optimal for FCP, such as ProRes?
This could also be a limitation in hardware. Are you loading the Maya material to a scratch disk that is external that has enough throughput to handle it, then importing into FCP? Where FCP is reading the files from could be giving you problems too, even though you don’t see a dropped frames warning. Also, keep your number of video tracks at a minimum, this eats up throughput.
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