I don't understand why the 1st and 4th records appear in the results. Does anyone understand why this happens based on the following code?

Because, if you use DST_AMT in the HAVING or WHERE clause,
VFP compare the original field DST_AMT,
and when APTRXDST have more of one rows not zero for one VOUCHER_NO,
then for equal SUM you have different comparison.

THIS IS THE CORRECT SQL SYNTAX

CREATE CURSOR APTRXFIL (VOUCHER_NO I,INV_AMT Y)
INSERT INTO APTRXFIL VALUES (76171 ,606.20)
INSERT INTO APTRXFIL VALUES (76172 ,301.87)
INSERT INTO APTRXFIL VALUES (76173 ,667.75)
INSERT INTO APTRXFIL VALUES (77093 ,706.14)

CREATE CURSOR APTRXDST (VOUCHER_NO I,DST_AMT Y)
INSERT INTO APTRXDST VALUES (76171 ,303.10)
INSERT INTO APTRXDST VALUES (76171 ,303.10)

INSERT INTO APTRXDST VALUES (76172 ,200.00)
INSERT INTO APTRXDST VALUES (76172 ,301.87)

INSERT INTO APTRXDST VALUES (76173 ,675.50)

INSERT INTO APTRXDST VALUES (77093 ,706.14)

SELECT ;
		APTRXFIL.VOUCHER_NO, ;
		APTRXFIL.INV_AMT, ;
		SUM(APTRXDST.DST_AMT) AS SUM_DST_AMT ;
	FROM ;
		APTRXFIL, ;
		APTRXDST ;
	WHERE ;
		APTRXFIL.VOUCHER_NO = APTRXDST.VOUCHER_NO ;
	GROUP BY ;
		APTRXFIL.VOUCHER_NO, ;
		APTRXFIL.INV_AMT ;
	HAVING APTRXFIL.INV_AMT <> SUM_DST_AMT  ;
	INTO CURSOR tmpAPTransactions
BROWSE

If you want preserve the name, then you must reuse the SUM() on HAVING

SELECT ;
		APTRXFIL.VOUCHER_NO, ;
		APTRXFIL.INV_AMT, ;
		SUM(APTRXDST.DST_AMT) AS DST_AMT ;
	FROM ;
		APTRXFIL, ;
		APTRXDST ;
	WHERE ;
		APTRXFIL.VOUCHER_NO = APTRXDST.VOUCHER_NO ;
	GROUP BY ;
		APTRXFIL.VOUCHER_NO, ;
		APTRXFIL.INV_AMT ;
	HAVING APTRXFIL.INV_AMT <> SUM(APTRXDST.DST_AMT)  ;
	INTO CURSOR tmpAPTransactions

next is ambiguous:

SELECT ;
		APTRXFIL.VOUCHER_NO, ;
		APTRXFIL.INV_AMT, ;
		SUM(APTRXDST.DST_AMT) AS DST_AMT ;
	FROM ;
		APTRXFIL, ;
		APTRXDST ;
	WHERE ;
		APTRXFIL.VOUCHER_NO = APTRXDST.VOUCHER_NO ;
	GROUP BY ;
		APTRXFIL.VOUCHER_NO, ;
		APTRXFIL.INV_AMT ;
	HAVING APTRXFIL.INV_AMT <> DST_AMT  ; && VFP use APTRXDST.DST_AMT
	INTO CURSOR tmpAPTransactions

Fabio