It may be helpful to keep in mind how data packets travel across the Internet. Basically everything has a destination "to" and source "from" IP address. The key to understanding where your email actually goes is found in the DNS servers. The DNS servers are pretty much like your local machine's hosts file. Each entry is a single line with two pieces of information. The first bit is a domain name and the second is the IP address associated with that domain name. Thus the sole purpose of DNS servers is to convert human readable addresses to IP addresses.
The hosting company will configure their DNS server to associate the RiversideFarms.com domain with a specific IP address that they own. Thus when you switch to a new e-mail provider, your existing domain name is assigned a different IP address, one that is owned by the new provider. A conflict will exist until the new IP address information propagates to all the DNS servers over the entire Internet.
While I can't say with certainty, I'm fairly confident that an error message would be generated telling the new provider that the domain they are trying to configure is already assigned an IP address. What happens at that point would be speculation on my part. The new people should not try to configure your old domain to their IP address without some action taking place by the old e-mail providers. That means the old e-mail providers must delete your domain from their DNS server first so that there are no network conflicts.
What about your actual mail? Your existing e-mail file content stays on the old server for a predetermined amount of time and/or until the disk space quota is filled. When you change service providers you most likely won't get a choice to move what is on the old server to the new server for reasons already explained. It's technically possible but not done in practice. Thus, whatever you do not download from the old server will be lost. The problem isn't going to be what you already have. There will be issues with incoming e-mail. When you make the transition to a new e-mail server, where will that new mail go? The answer is that it depends on what the network DNS servers have in their routing tables. The new provider will update the DNS information on their end, but it can take a week for that new information to propagate to the DNS servers all over the world. Anything sent to you during that week may or may not arrive. That would be the reason to use a different domain name, or to deal with the possibility of some mail not being delivered. Of course you could send e-mail immediately without conflict. The headers in anything you send will contain IP information of the server from which it was sent.
We are talking e-mail here, but the same rules apply to a website being hosted on somebody's server. In the case of a website you may not have access to it until all the appropriate DNS servers have updated. That exact scenario has happened to this website in the past. It only took overnight for the Internet to update, and I suspect it would be the same with e-mail. A lot depends on the phase of the moon.
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