Had an issue about two weeks back with our Moch's (loose, slides up and down like described in OP) band getting caught on one end of the sliding gate in her smaller cage (that you lift up to remove the food and water trays).
Mochi's band:
Fortunately, we were right there and noticed when she seemed "stuck" on the side of her cage and began flapping her wings furiously. She was able to unhook the caught limb without any apparent injury, but I figure we would've used two sets of hands to hold her still and jimmy it off if the bird hadn't been able to safely free itself.
Used GF's jewelry pliers to 'bend' the offending part further inward, and do the same for all the other closures/loops around the cage. It's a plastic-on-metal lighter type of cage - inexpensive and easy to travel with. Steel welded-metal cages don't have this particular issue, but if they did, you probably couldn't fold in the loop with a pair of pliers. No cage is perfect, anyway -welded metal ones (we have also have the
Prevue Hendryx Wrought Iron Select) can sometimes have rough, un-grinded-down burrs.
I actually went and bought a pair of specialized band-clippers from here:
Lady Gouldian Finch .com - Leg Bands
Alternate Site with same/similar clippers
3rd Site w/ the same ones
that works for birds up to cockatiel sized (GCC being smaller or approximately same, so suitable). I haven't used them - they're a sort of 'break glass' kind of item because Mochi hasn't since or before had any incidents, and even with two sets of hands it can be hard to safely stabilize a squirming bird for band removal. But definitely don't want a broken/sprained leg, so that's what they're there for, and we may try to use them sometime in the future. They have curved edges and are designed to cut the band cleanly off with two strokes on opposing sides of the band, without bringing the blades' tips into contact with the bird's body. They work on aluminum bands (so Mochi's plastic band should give no resistance).
I did find this video online of someone using very similar clippers to remove a band from a Barred Parakeet:
https://youtu.be/pm-QhxwuuwY?t=55s
Interestingly, her clippers have nice rubberized grips and more greatly curved edges, which would have been nice to find online somewhere,
but she provided no link (beyond 'Google bird band cutters') 
Actually, Googling 'bird leg band cutters'
does show it.
EDIT: Read through the YT Comments, she got her clipper tool through the AACC (Avicultural Advancement Council of Canada) bird club in her area. No direct link or mention on their site, but they do have
a banding order form, so you could contact them if you really wanted those exact ones.
EDIT: I found the exact ones used in the video, with the red grips:
Band Cutter #2 | NFSS | National Finch & Softbill Society