Christ did not come to destroy the law but to magnify it. He magnified the law from the letter of the law to the spirit of the law by saying to lust after a woman in one's mind is equivalent to adultery. But in magnifying the law how can the Sabbath be moved from one day to the next when Christ himself said to keep the commandments?
The reference to lust and adultery is found in Matthew 5:27-28, but I would like to focus on your allusion to Christ's mission that is found in Matthew 5:17-18: "Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill. For assuredly, I say to you, till heaven and earth pass away, one jot or one tittle will by no means pass from the law till all is fulfilled." Notice that you altered the word "fulfill" to "magnify."
The word "fulfill" comes from the Greek word pleroo, which means "fill, make full, supply fully, complete." Its meaning can be seen in how it is used in other passages. For example, there are numerous passages that speak of something being done in order that a prophecy might be fulfilled, such as in Matthew 1:22, 2:15, 17, etc. The word means that the prophecy was answered in full and brought to completion. When something is made completely full by a task, the task is finished. We say this in English when we say we went to the gas station and filled up the car. In other words, we stopped pumping gas into the car once it reaches the full mark because no more gas could be added. This sense of a completed purpose is seen in Luke 7:1, "Now when He concluded all His sayings in the hearing of the people, He entered Capernaum." The word "concluded" is the Greek word pleroo. Jesus fulfilled his purpose in that particular lesson. There was nothing more that needed to be said at that moment, so he stopped. It is also seen in Acts 19:21, "When these things were accomplished, Paul purposed in the Spirit, when he had passed through Macedonia and Achaia, to go to Jerusalem, saying, "After I have been there, I must also see Rome."" Paul had completed what he set out to do in Ephesus, so he looked to move on to another region. The same Greek word pleroo is being used, though it is translated as "accomplished."
Jesus explained what he meant by "fulfilled" when he emphasized that the Law would not end until all was fulfilled. This "fulfilled" comes from a different Greek word, ginotai. It means, "to be, to come into being, to be made, be done, become, or to be celebrated." In other words, Jesus is saying that the world would have a better chance of ending before God accomplished His purpose for the Law.
Another verse that speaks to this same topic is Romans 10:4, "For Christ is the end of the law for righteousness to everyone who believes." The word translated "end" is from the Greek word "telos," which means "end, termination, conclusion, aim, result, goal, outcome." In other words, the purpose of the Law culminated in Christ Jesus. Jesus was its goal. With His death, he brought the law to its conclusion. "Having wiped out the handwriting of requirements that was against us, which was contrary to us. And He has taken it out of the way, having nailed it to the cross" (Colossians 2:14). Or as Paul said in another letter, "Therefore the law was our tutor to bring us to Christ, that we might be justified by faith. But after faith has come, we are no longer under a tutor" (Galatians 3:24-25).
But in completing the purpose of the Law, I don't want to leave the impression that Jesus left us lawless. "Not that we are adequate in ourselves to consider anything as coming from ourselves, but our adequacy is from God, who also made us adequate as servants of a new covenant, not of the letter but of the Spirit; for the letter kills, but the Spirit gives life. But if the ministry of death, in letters engraved on stones, came with glory, so that the sons of Israel could not look intently at the face of Moses because of the glory of his face, fading as it was, how will the ministry of the Spirit fail to be even more with glory? For if the ministry of condemnation has glory, much more does the ministry of righteousness abound in glory. For indeed what had glory, in this case has no glory because of the glory that surpasses it. For if that which fades away was with glory, much more that which remains is in glory" (II Corinthians 3:5-11). The glory of the old Law was fading, but what remains is more glorious. The Hebrew writer put it this way, "But now He has obtained a more excellent ministry, inasmuch as He is also Mediator of a better covenant, which was established on better promises" (Hebrews 8:6). But also notice that Paul said that what remained was a new covenant. "In that He says, "A new covenant," He has made the first obsolete. Now what is becoming obsolete and growing old is ready to vanish away" (Hebrews 10:13).
Now just in case you are inclined to argue that the Ten Commandments were not apart of the covenant: "So He declared to you His covenant which He commanded you to perform, the Ten Commandments; and He wrote them on two tablets of stone" (Deuteronomy 4:13).
No one "moved" the Sabbath. It's purpose, along with the rest of the Old Law, was brought to an end and replaced with something more glorious.