[Fizzlemizz]

Bag 0 is the backpack so checking there first for empty slots is probably not what you're looking for.

for bag = 1, 4 do

Edit:
In a tight loop, the bag inventory doesn't update in time so you the get the "That bag is full" notification which can cause the process to break. Best to wait for the bag update event before processing the next item:

local LastBag, LastSlot, Remove
local function SaveToBags()
    while Remove do
        PickupInventoryItem(Remove)
        Remove = Remove+1
        if Remove > INVSLOT_LAST_EQUIPPED then
            Remove = nil
            return
        end
        if CursorHasItem() then -- only process picked up items
            local Placed
            for bag = LastBag, 4 do -- bag 0 is the backpack
                local slots = GetContainerNumSlots(bag)
                for LastSlot = LastSlot, slots do
                    local itemID = GetContainerItemID(bag, LastSlot)
                    if not itemID then
                        PutItemInBag(bag+19)
                        Placed = true -- so scram
                        break
                    end
                end
                if bag > LastBag then
                    LastBag = bag
                    LastSlot = 1
                end
                if Placed then break end
            end
            if not Placed then -- no empty slots in bags
                local GotSpace
                for i=1, GetContainerNumSlots(0) do -- make sure the backpack isn't full as well
                    if not GetContainerItemID(0, i) then
                        GotSpace = true
                        break
                    end
                end
                if not GotSpace then  -- No more slots in backpack to put stuff either
                    print("Not enough backpack space!")
                    Remove = nil
                    return 
                end
                PutItemInBackpack()
            end
            break
        end
    end
end
 
local f = CreateFrame("Frame")
f:RegisterEvent("BAG_UPDATE")
f:SetScript("OnEvent", function(self, event, ...)
    if event == "BAG_UPDATE" then -- if you are using the handler for more than this event
        if Remove then
            SaveToBags()
            return -- maybe, if you are doing other things with this event as well
        end
--  elseif event == "SOME_EVENT" then
    end
end)
 
SLASH_XXX1 = "/xx"
SlashCmdList.XXX = function()
    LastBag = 1 
    LastSlot = 1
    Remove = INVSLOT_FIRST_EQUIPPED
    SaveToBags()
end

Edit: I didn't notice Kanegasi had made changes before doing this...