I'm not sure how much money you are willing to spend to do this...
But you could try the following...
However many fish you have...according to size..place each in a tupperware container (or anything comparable that actually has a snap on lid (that forms a seal)..that will not come off..those made for shoes that cost $.77 at Walmart WILL NOT work ..the lids come off way to easy. I think Ziplock has some that are quite affordable though..comes in a 4pack and also avliable in several different sizes).
You can then get one of the large storage bins from Walmart (they have some on sale right now, 22 gallon size for $4 if you don't have any free) and stack them inside the storage tub.
I would remove the gravel from the tanks..as it can and will break through the bottom of your tanks when you try to transport them with it in them..I've had it happen...and it SUCKS lol..since #1. it's heavy to begin with and #2.. it's wet.
That too can be placed in one of the storage tubs..with tank water....if you are worried about your beneficial bacteria dying..you can buy a battery operated air pump (or get one of the jacks that has a electrical plug outlet...that plugs into your cars cigarette lighter if you don't want to buy a pump you probably won't ever use again) and run an air line inside it...that should keep the bacteria from dying from lack of oxygen. I would place my filter pads inside this as well.
Also, make sure to keep putting the water you change from teh containers in there..so it has something to live on too.
If you cannot immediantly set your tanks back up..you can also buy airline and a few T connectors and make a system to supply your fish with air. I did this when I had to change out my gravel in my community tank (until the tank was cycled again..since I replaced all my gravel).
A regular size air pump will work on approx. 3 boxes. I had 6 containers so I had to use 2 air pumps (they are the small air pumps..not larger ones though..the cheapies).
I drilled a hole just big enough for the air line to fit in the lid of each container (I would do this after you transport the fish..not before..or the water could splash out).
I also put some of the plants from the tank in each container, to make the fish feel more comfortable and less stressed, so they would atleast be able to associate with something around them and not feel so alienated. Also keeps gives them a place to get away from each other, if they need to, so they aren't just in each others face the whole time.
If you fish need certain temps...you might want to get a heating pad and sit part of each container on it..so they don't get too cold too.
This is how I had mine....
I had mine in the .77 cent shoes boxes..but I wan't moving either.
Each container held 1.75 gallons of water, which I had to change daily.
1 container....4 rasboras
1 container...4 pristella tetras
1 container...2 cories, and 1 neon
1 container...1 UDC
1 container....snails
1 container... for holding each group during water changes
I drilled the holes in the lids using a Dremmel and the circular metal bit. But a regular drill bit/drill would work just as well. Might coudl even use a cordless scrwewdriver.