Is it possible to do back-to-back broadening assignments?

While it is true ( as others have stated) however it is not likely that a Branch Manager would approve it. The word broadening is only broadening as long as the service emeber returns to their career field and applies such skill sets as developed during the time away.

If we are talking career here I can unequivocally say with a few accepting, selection to the next grade by a board will look more favorable at a successful broadening tour and return home than they will for someone gone for six plus years even if successful. Having said that, if you excell at both and are rated above your peers and set sone sort of example as to be rated appropriatley then there is the acception.

So, only you know your capabilities.

Is is 'possible'? Absolutely. Is it 'likely'? Depends.

This is a conversation you'll have to have with your assignment manager*. It's all going to depend on what shortages are occurring at your grade. Suggest you give them a call and discuss with them.
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* https://www.hrc.army.mil/content/Enlisted%20Infantry%20Branch%20POCs

Possible? Sure. Depends on what your Branch says. You may run the risk of missing "time on the line" which could bite your six. Talk with Branch, get their input and go from there. IMHO, do recruiting, then go back to the line for a couple of years, then go DS.

Maybe if that's what you want. However it use to be frown upon when it came time to be selected for E8 and E9. Times may have changed over the years.

In assuming it is similar with the Marine Corps you might be able to thread the needle. There is a slim chance the Drill field is hurting for bodies with a priority that could help. But generally you will run into some static trying to do those two duties back to back. That's a lot of time out of your MOS. Especially if you extend on either which they normally offer at some point. Do one and get some MOS time for 2 years in-between. It could hurt you in the log run when you go back to your MOS with that much away time.

A Friend Was DA Select For Recruiting As A SSG. Did His Tour , Then Volunteered For Drill He Made SFC During His Tour At Drill, He Then Went Back To The Line As An 88m , He Retired As A SFC, He Was Told He Missed Too Much Time On The Line To Get Promoted To MSG .
That Was About 10 Years Ago