Definitely transport fish in suitable bags - most lfs will provide these at minimal/no cost - make sure you can securely tie off bags with elastic bands (again lfs usually have proper stretchy bands)
As you’ve only a few fish/bags, take the time to tape off the bag corners (unless your lfs has those excellent fish transport bags)
Fill bags 1/3 water, 2/3 air (unless you manage to find “breathable” bags - these are much more expensive and seem less common now)
Stack bags securely in a styrofoam box (or similar) so that temperatures remain more consistent (again lfs often have styrofoam shipping boxes in need of rehoming), close the lid and keep fish in quiet darkness until you’re able to transfer them to temporary bins - fish will be fine overnight with basic sponge filters, heaters, and extra doses of Prime (or Stability)
(If bagged and stored appropriately etc, fish can easily manage 48-72h)
For your current filters, try to clean/rinse media in the weeks up to the move, then you can just drain filters and transport - as long as they’re kept cool rather than hot, with air (oxygen) access, damp/humid, the biological media seems to manage very well
(I leave a couple cm’s water in the bottom of the filter (below the media basket), and make sure the “taps” are open (may also have the top slightly ajar if this will be longer than 24h))
Definitely transport Corydoras separately - also double bag (ask lfs for a demonstration if you’re unsure); I’d also spread Cories over a few bags
While it’s not common to lose Corydoras during bagging/transport, it does happen (and no, chasing the Corydoras about first, before netting, did not prevent subsequent deaths (due to assumed poisoning - this seems to be more prevalent with certain species, also mature rather than juvenile fish)
While transporting in buckets may “seem” beneficial, fish are actually subjected to more physical stress (from the water movement) and often reduced oxygen/air exchange (relative to suitable bagging)
When you arrive at your new home, just leave the fish in a quiet relatively warm corner where they won’t be stressed by moving boxes and people
When things have quieted, prepare your temporary bins, check water temp and parameters (compared to previous tap - you should have a good idea before the move, if water will be similar), make sure sponge filters/filters/heaters etc are all working properly ... then begin unboxing fish
As for temporary bin color, there’s a paper on that somewhere - fish did find some colors less stressful than others - and blue was not the “best” color (unfortunately I don’t recall the exact order any longer), choose a food grade plastic bin in a dull dark color (green blue black brown)
Of course a lighter translucent bin is better for monitoring fish so I often use this instead, just adding a blue or black background to 2 or 3 sides depending on fish species and bin location
You can also add floating plants, and some wood etc for hiding areas (I like to suction the wood to the bottom/walls so it can’t shift about)