διά is a highly flexible preposition whose meaning depends on the case of the noun that governs it.
It is often used in a perfective sense. Hebrews 1:11, “they perish, but you abide through (διαμένω).”
In composition, it expresses “thorough”, “between”, and “through.” Passing through Macedonia (1 Corinthians 16:5). Some of the scribes were thoroughly reasoning in their hearts (Mark 2:6). Collecting only what is through appointment for you (Luke 3:13). διαθήκη conveys a covenant—established between (Galatians 3:15).
When used with the Genitive case, διά expresses the intermediate agency through, indicating the means by which something is achieved. God sent His Son so that the world, through Him, should be saved (John 3:17). The word of the Lord, spoken through the prophet, was fulfilled (Matthew 2:15).
διά with the accusative case conveys “because of” or “for the sake of.” In Matthew 6:25, Διὰ τοῦτο expresses “because of this.” This combination is used about 140 times in the New Testament, conveying “on account of”, “due to”, or “as a result.” Christ was delivered over (betrayed) because of our trespasses and was raised because of our justification (Romans 4:25). In Mark 2:27, διά expresses the purpose, benefit, or advantage. “The sabbath came to be for the sake of man, not the man because of the sabbath.” Abraham’s faith being counted to him as righteousness was not written for his sake only, but also for our sake (Romans 4:23–24).
διά can convey agency (by, through). In 2 Corinthians 1:19, the Son of God who was proclaimed by (διά) us. Paul’s appointment as an apostle was not by (διά) man, but by (διά) Jesus Christ and God the Father (Galatians 1:1).
διά may be used to express “means.” He spoke by means of a parable (Luke 8:4). The Lord spoke to Paul in the night by means of a vision (Acts 18:9). When διά is used with τί, it expresses “why.” When asked by Jesus where John’s immersion was from, the Pharisees reasoned among themselves that if they said it was from heaven, then Jesus would ask why διά τί they did not believe him. Why (διά τί) did Israel not obtain righteousness? Because they did not seek it out from faith, Romans 9:32. This combination is used around 52 times in the New Testament.
There are a few cases where διά expresses “by” in Koine writings: A writer sending two letters by two men, and a document that is to be signed by (ref. G. Milligan’s Greek Papyri, pages 39 and 40). Although διά may be used to express agency, it does not hold the same strength as ὑπό. Thus, διά conveys that Jesus is the intermediate agent by which creation was made, not the sole agent (Hebrews 1:2).