The same throwing-out that frees the captive can describe being slandered, driven from the synagogue, or finally shut out. Casting out demons is no proof of belonging — and the verb can run the other way.
τῷ σῷ ὀνόματι δαιμόνια ἐξεβάλομεν;
daimonia exebalomen
"Did we not cast out demons in Your name?" … "I never knew you."
Casting-out is not the same as being known by Christ. Mighty works without relationship are nothing.
οὐκ ἐκβάλλει τὰ δαιμόνια εἰ μὴ ἐν τῷ Βεελζεβούλ
ouk ekballei … Beelzeboul
This man casts out demons only by Beelzebul, the ruler of the demons.
The slander that prompts the "blasphemy of the Spirit" warning: attributing Spirit-wrought deliverance to the enemy.
καὶ ἐξέβαλον αὐτὸν ἔξω
kai exebalon auton exō
And they cast him out … Jesus heard, and found him.
The healed man is thrown out of the synagogue — and immediately sought and found by Jesus. Cast out by men, taken in by Christ.
οἱ … υἱοὶ τῆς βασιλείας ἐκβληθήσονται εἰς τὸ σκότος
huioi … ekblēthēsontai
The sons of the kingdom will be cast out into the outer darkness.
The gravest use: a passive and future form — final exclusion. The whole point of 6:37 is that those who come to Him are exempt from this.