gnu make is a little hard to understand sometimes... this is telling in regards to how it handles environment variables generally, within rules and within sub makes

make.stdout

root.ruleA You ain't got no kind of feeling inside
export TMP="I got something that'll sho nuff set yo' stuff on fire"
make -C ./submake
make[1]: Entering directory '/home/ismael/Fiddles/make-bug/submake'
child You ain't got no kind of feeling inside
make[1]: Leaving directory '/home/ismael/Fiddles/make-bug/submake'
root.ruleB You ain't got no kind of feeling inside
make -C ./submake
make[1]: Entering directory '/home/ismael/Fiddles/make-bug/submake'
child You ain't got no kind of feeling inside
make[1]: Leaving directory '/home/ismael/Fiddles/make-bug/submake'
root.ruleC You ain't got no kind of feeling inside
make -C ./submake TMP="What I got will knock your pride aside"
make[1]: Entering directory '/home/ismael/Fiddles/make-bug/submake'
child What I got will knock your pride aside
make[1]: Leaving directory '/home/ismael/Fiddles/make-bug/submake'
root.ruleC You ain't got no kind of feeling inside
make -C ./submake
make[1]: Entering directory '/home/ismael/Fiddles/make-bug/submake'
child Tell me something good
make[1]: Leaving directory '/home/ismael/Fiddles/make-bug/submake'

Makefile

export TMP="You ain't got no kind of feeling inside"

.PHONY: all

all: ruleA ruleB ruleC ruleD

ruleA:
	@echo root.ruleA ${TMP}
	export TMP="I got something that'll sho nuff set yo' stuff on fire"
	$(MAKE) -C ./submake

ruleB:
	@echo root.ruleB ${TMP}
	$(shell export TMP="You refuse to put anything before your pride")
	$(MAKE) -C ./submake

ruleC:
	@echo root.ruleC ${TMP}
	$(MAKE) -C ./submake TMP="What I got will knock your pride aside"

ruleD:
	@echo root.ruleC ${TMP}
	$(eval export TMP="Tell me something good")
	$(MAKE) -C ./submake

submake_Makefile

# Note: this should be submake/Makefile, but gisthub doesn't support subdirectories
.PHONY: all

all: child

child:
	@echo child ${TMP}