Do these work?
in AM16 of June sheet:
=IFERROR(INDEX(REMARKS,MATCH(1,INDEX((FULL_NAME=$B 16)*(tDATA[DATE OF TRANSFER OUT/DROP OUT]<=$AC$6),0),0)),"")
copied down
In B16 of July sheet:
=IFERROR(INDEX(FULL_NAME,MATCH(SMALL(IF((COUNTIF($ B$15:$B15,FULL_NAME)=0)*(GRADE=$AC$8)*(SECTION=$AI $8)*(DATE_ENROLLED<=AC$6)*((tDATA[DATE OF TRANSFER OUT/DROP OUT]="")+(tDATA[DATE OF TRANSFER OUT/DROP OUT]>EOMONTH(AC$6,-1)))*(SEX="M"),COUNTIF(FULL_NAME,"<"&FULL_NAME),"" ),1),COUNTIF(FULL_NAME,"<"&FULL_NAME),0)),"")
confirmed with CTRL+SHIFT+ENTER and copied down
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