switching between web and email mid-process is bad and wrong

 You're doing something on someone's web site, and it actually seems to be working for once.  Then, suddenly, it tells you to check your email to continue.  You are no longer doing something on someone's web site.  You are trying to switch, from web to email.  You have to access whatever email address it is this web site tried sending something to, to continue the process.

The problem is: email can take a long time to arrive.  The user may access it in a completely different way: on a different device, a different computer, at a different location or time.  It may never arrive.  There are many glitches with email, as it's flawed by design.  Many messages just never get thru, and are filtered out at some indeterminate point on their journey.  Often, a message is misclassified as spam, and ends up in the wrong folder.  Various user-agent misconfigurations can result in a message not being seen.

By switching from the web to email half way thru a process, the service provider is switching from something known to be working, because the session so far is in the browser, to something whose workingness is unknown.

Even when everything is working great, it's annoying for the user, to be made to switch.  Notifications by email with important information, as long as they are limited in quantity, are nice, because the information persists.  But this doesn't mean switching the process from web to email.  Things can continue on the web, with emails fired out sideways as and when.

An especially annoying class of this is when required information is sent by email, as a result of a web interaction, and the web side of it does not even tell the user about the email bit.  The user just gets to find out next time they read their email.  Or perhaps they are looking for the specific information on the page, and it's not there, and they guess that maybe they have something waiting for them in their email.

Take this example:

I need my booking reference number to complete the Passenger Locator Form.  Where is the reference number?  I try calling and sit on hold.  They know I need the reference number.  Where is the reference number?  Is it in that QR code?  No, that's a link to their app, FFS.  Let's read it a few more times, cos everyone needs the reference number, they can't just not put in the reference number.

Eventually I guess: maybe it's waiting in an email.  And it was.

If you're designing a service, don't do this.  Put all the info in the web page, even if you are sending the same info by email.