The theme of "The Moving Finger" is generally regarded as relating to control. Ralph's first wife controls him so thoroughly that he is compared to the host for a parasite. He loses himself entirely; he isn't even aware of himself until she passes.
When Ralph marries the second Mrs. Grancy, he lives his fullest life through his happiness with her. If he is to be believed, the relationship was truly mutual, and they were both so overcome with joy at their union that their unexpected and sudden separation when she died was almost incomprehensible to Ralph Grancy. He realizes, through her continued "presence" in his life (it is not clear in the story if this presence was real or imagined), that he needs to bring her along, so to speak, into old age so that she will not be alone without him, even in death. This is why Ralph asks Claydon to age the masterpiece he painted of Mrs. Grancy—so that the couple can, in a way, grow old together.
Claydon believes himself to be in love with Mrs. Grancy, though the reader is given no clear indication that those feelings were ever returned. What both men share is a love for this woman that becomes obsession. Whether there is a paranormal occurrence that allows Mrs. Grancy to take hold of these men in the afterlife, or their fantasies grip them to the point of delusion, Mrs. Grancy has a firm hold on both men. Claydon uses his innate knowledge of Mrs. Grancy, and Ralph's desire to continue experiencing his life with her, to nudge him into death with merely a look upon the portrait's face. This is how Claydon comes to own Mrs. Grancy (the portrait) once and for all, remaking her back into his ideal image of her.
This control is nuanced; neither man is trying to control or manipulate Mrs. Grancy as she was when living. They have literally objectified her, projecting all their fantasies onto a portrait. Ralph Grancy idealizes unyielding companionship. Claydon idealizes the point in a woman's beauty when youth has ripened into womanhood, where innocence and experience exist in balance.
Both men want so badly to seize onto how Mrs. Grancy made them feel in those moments that they wanted her frozen at those moments in time—not because they desire to own the woman, but because they feel compelled to own those fleeting moments when they saw themselves idealized in her. They were chasing their own reflections in her face, like wanting to catch time in a jar.