Does a domain's Glue Records get transferred when I transfer the domain to another registrar?

It depends on the registrar and on who's providing the DNS servers for DOMAIN.NET

In your example .NET is the top-level domain. Omitting some details, when a computer on the Internet is trying to get to a computer in DOMAIN.NET

• it already knows how to find the nameservers for .NET (and the other top-level domains like .COM .ORG .CA and so on), so it goes to one of those and asks for the address of the nameservers for DOMAIN.NET
• the nameservers for .NET know the IP addresses of the nameservers for DOMAIN.NET (those are the glue records)
• and then the nameservers for DOMAIN.NET know the IP addresses of computers within your domain

In principle, if you are running your own DNS servers and simply change registrars, the glue records don't change and you could probably switch by doing nothing.

The way your question is worded, it sounds like the registrar is also providing the DNS servers for your domain, in which case the glue records don't need to be transferred, they need to be changed, and that's not going to happen automatically.

If you're relying on your registrar's name servers, you need to talk to the new registrar and find out when they'll change the glue records after you transfer the domain to them. Then you need to talk to the old registrar and make sure they continue to provide services (with the old glue records) until the change takes place.