Well, this is what USCIS says for travel during pending application:
You may travel if you are in valid H-4 status and meet all the admission requirements, including having a valid H-4 nonimmigrant visa.
However, traveling outside of the United States could cause delays in your case. While you are outside of the United States, we may need additional information to make a decision on your Form I-765 or we may issue a Notice of Intent to Deny (NOID) with an opportunity to respond.
If you do not respond on time to a Request for Evidence (RFE) or to the NOID, we may deny your case as abandoned.
Additionally, travel outside of the United States may also cause possible delays if we need to reschedule your appointment at an Application Support Center.
In simple terms, I would travel outside if someone in USA (like your H1B spouse) can receive the notices from USCIS and respond, if required.
In a normal scenario, you do not receive RFE or need to visit application support center.
But, if you do, you will have to come back to USA.
Is Stamping required to re-enter USA if H4 visa has expired?
Yes, you need VALID visa stamp to re-enter USA with the exception of automatic visa re-validation available from Canada and Mexico border (if you come back within 30 days).
How Automatic Visa Re-Validation works and documents required:
I read your today post on H4EAD last stage of drafting the rule. Will there be any quick response expected on this case?
What will be the expected outcome? What will happen to existing H4EAD?
What will happen to applied H4EAD?
After a long wait I filed H4EAD last week, Monday Aug 13 2018. When I will get further communications like receipt number? We are in the confused stage at this moment. As I requested in the last post about Canada travel I about to take graduate certificates program from Canada. I believe that will help me to establish job hunt and PR process in Canada. On the other hand after a long wait i applied H4EAD. I don’t know how to take decision at this moment to move forward. I am 12 years of experience with MASTER DEGREE (not from USA) at this moment.
I have to stay inside USA to get H4EAD to start my career in USA or I shall travel to Canada to start my course? (Fall term starting sep2018)
I need your opinion here that will allow me to think out of box.
You can opt for Automatic visa revalidation if your answer is YES to ALL of these questions
You have approved petition (i797 Approval) from USCIS (in your case unexpired and valid H4 i797 approval).
You have un-expired i94.
Your current visa stamp in your passport has expired.
You do not have any PENDING (or rejected) visa application with US embassy in Canada.
Visa Stamping in Canada
If you intend to get your H4 visa stamp in Canada, then, yes, you will have to fill form DS-160 (it is an application form for visa at US embassy).
If you do apply for Visa stamping in Canada, you CANNOT use the automatic visa revalidation to enter USA.
Your visa application at US embassy in Canada has to reach a result.
Your entry to USA will be defined by the result you get.
If it is pending, you still CANNOT use visa revalidation.
You have to wait in Canada to see the visa result and then proceed according to the result.