Generally, it's possible for a L visa holder to petition for a change of status from L1 to H4 while in the United States. Note, however, that the person must document to the USCIS that he is maintaining lawful status in the U.S., meaning they must have recent paystubs for their L1 employment. If the USCIS isn't satisfied with evidence of continued status, they may require that the L1 holder go to their home country to obtain the visa. Further, it's not possible to request premium processing, so this can be a protracted application process. For these reasons, for attempts to change status from L1 to H4 while remaining in the U.S., I would suggest consulting an immigration lawyer about the specifics of your case before deciding how to proceed.

Once you change your status to H4, you aren't eligible to work. You may remain eligible to reenter the nation on L1 status, if you leave, as you can simultaneously hold L1 and H4 visas, but you can only use one at a time.

Any dependents whose status is derivative of the L1 visa would want their status changed, as once the L1 visa ends so would any derivative visas.