Any thoughts H4 minor dependent can travel while their H4 extension ( I-539) processing in US . I mean whether consulates guys check I-539 approval while taking decision for H4 visa stamp if I apply from a US consulate in India ? Principle applicant has a approved I-797.
This should be okay. Generally, USCIS do not consider extension of status application as abandoned if the beneficiary travels outside the US. However if the beneficiary is not present in the US at the time USCIS adjudicates the case, they may deny it. You can still enter the US based on stampes visa and CBP will issue a new I94.
If your application is approved after you travel back to the US, you may end up with two I-94 ( one issued by CBP based on your visa stamp and another attached to the I797 approval notice). This is again not an issue as your status will still be valid and will be governed by the I94 attached to the I797 approval notice.
The mismatch in I94 can be fixed by travelling to Mexico/Canada border and entering back to the US on the I797 approval notice for extension of status.