How do they decide how much to allocate to their account(s)?
Is money easily moved around? Are there multiple signatories?
Yes, they generally spread across many banks, many accounts and even nations. Generally after $50M in revenue and for sure after $100M they cease having "standard" accounts like SMB's have. Honestly, most banks once you cross even the 10-20M annually will customize accounts for you almost anyway you want. Smaller the bank you use the smaller the revenue before they customize. Also insurance gets interesting insuring large sums of money too.
Allocations are handled via discussions with the banks, outside advisor groups and electronic systems. Usually the CFO of large organizations will employ a staff that literally manages investments and they will differ from the people managing daily/monthly/quarterly cash flow. There are literally teams of people that work internally and with banks, accounting firms, attorneys and other consultants to figure out the right allocation and distribution. It also gets super complex because of tax laws, international transactions, international revenue etc. So there are teams of people usually managing this not just one or two people. In one of the largest public firms I was at they had about 100 people that did nothing but manage investments and allocation, that team worked with the cash flow team on a quarterly basis and with the M&A team as necessary.
Signatory is not handled the same way as you would see in a SMB either, where you can say two people have to sign every check etc. In most large organizations everything is managed via the back office finance systems which will digitally sign everything, handle ACH/wire transfers etc. And those systems have built in purchase limits for different levels of approvals etc. I have seen where an ESVP could spend $5M in one transaction basically unquestioned, but the reality is they'd never really do that without discussions. But it is there so they could make emergency purchases of equipment, supplies etc without need to have delays in the system. I even saw where the Admin Assistant for one of the CEOs I worked for had purchase approval of up to $50k a month, just to handle incidentals, travel, meetings, food etc without having unnecessary headaches put in her way.
Larger banks also seem to make it easier to remove personal guarantees on credit facilities as tbe company grows. As for how much to allocate to each account, most banks provide a sweep service if balances are large enough where the balance of each account is sweeped nightly into a money market account, and then restored the next day, Otherwise you just need to maintain balances in each bank account in support of your cash flow projections.
As for international finance, I have found that foreign banks (e.g. ANZ Banking Group, Royal Bank of Canada) are much better with transfers and moving money around than US banks (as they are not stymied by the Federal Reserve and the Fed Payments System). Do not overlook Bitcoin either.
Where small local banks shine for us is through their trust departments. Larger banks tend to have trust products and packages for "wealth management" into which funds are dubiously recruited, while with small trust companies, you can engineer boutique situations (e.g. where fiscal instruments such as securities and time deposits can be held as collateral but still traded on the open market under a trust account). Befriend a small bank trust officer!
Transfer of large amounts of money between banks in the US is still very much a credit transaction - availability of funds are determined by the credit of the parties to the transaction. New companies with large transfers even between large banks will get restricted funds availability, until credit develops. Working with larger banks, it is much easier to establish funds availability in conjunction with a more holistic credit picture.
I'm sure for legitimately large corporations, there is a need to diversify among the top 13 banks, especially for national businesses (e.g. not even Wells Fargo is in Missouri or Oklahoma). We have found there effectively are NO national banks in the USA present in all states (we have personnel in all the lower 48). As in politics - when you're small you are either Democratic or Republican. When your large it is necessary to contribute equally to both. I think the same is true of banks - a business of $100M or more would want to diversify even if just for risk management.
I agree with @davismwfl that you need to construct a banking relationship around your systems.