I've heard of some area where the dispatchers are technically considered LEOs and sometimes end up doing stuff like assisting with searching incoming prisoners and such.

It also kind of highlights another issue with the field, on the federal level we're classified as clerical staff, basically just basic office staff. Different states, counties, and such may make different classifications, but it's a bit of a crapshoot. That can have an effect on what kind of benefits we're entitled to, how much we get paid, when we can retire, etc. There's a bit of a push to have us reclassified to the same category (I believe it's called "protective services" or something along those lines) as first responders (I believe the most recent version of that bill was called the 911 SAVES act, not sure what the current status of that is.) One of our little propaganda slogans is that we're the first first responders. I don't necessarily think we should be in the same category as cops and firefighters and such, were not on location putting ourselves physically in danger, but the kinds of stress and such from the job are probably more in-line with what they do than what some clerk working in an office at the courthouse experiences, IMO a 3rd category is probably most appropriate, but it's probably a lot easier to just reclass us as first responders than to creat a whole new classification from scratch.

And depending on where you work, dispatch may be part of the local PD, it may be part of the county sheriff's department or state police, or it may be part of your county's department of public safety (that's the case where I work)