How does the relationship between Marianne and Colonel Brandon change over the course of the novel?

While it is true to say that Marianne has no attraction toward Colonel Brandon because he is too old and wears "flannel" waistcoats, it is not true to say that, as a consequence, she rejects his suit: Colonel Brandon makes no suit for her love--thus no suit can be rejected--until well after Edward and Elinor are married and settled in the parsonage at Brandon's estate of Delaford. It is equally untrue to say that at any point Brandon continues a courtship of Marianne: Colonel Brandon never has the heart to initiate a courtship until Marianne spends considerable time in visits to Elinor and Edward at Delaford, and then only with Mrs. Dashwood's assistance and encouragement.
Marianne Meets Brandon
How does Marianne meet Colonel Brandon and on what is her long-lasting first impression formed? Marianne and all the Dashwoods are invited to take dinner at Barton Manor. It is at this dinner that Marianne meets Colonel Brandon. Her first assessment of him is that he, at thirty-five, is "old," "infirm" and in "declining life." She finds a modicum of respect for him because he is attentive, although not rapturous, about the music she sings and plays. She considers him an unfortunate man of "advanced years" in a "forlorn condition as an old bachelor." So, had he been, in her opinion, pretentious enough to advance a suit of marriage, she indeed would have rejected it. Yet pursuing a courtship of Marianne was not a possibility Brandon even contemplated because of Marianne's decided disinterest in him.
Second Attachments and Flannel
Although, as Mrs. Jennings and Sir John noticed, Colonel Brandon had an immediate "partiality" for Marianne (we later learn the immediate nature of his partiality was due to her striking resemblance to Eliza), Marianne has, as shown above, an immediate disdain for Brandon: Marianne was "prejudiced against him." On top of this dislike for thirty-five-year-old Brandon, Marianne was completely enamored of twenty-five-year-old Willoughby: "what could a silent man of five and thirty hope, when opposed to a very lively one of five and twenty?" In view of Marianne's sensibilities, it would take a bold and audacious man to pursue Marianne's affection, which we know Brandon was not; he was grave and "reserved."
In addition, Marianne's "romantic" ideas about love and attachments prohibit the existence of "second attachments," second loves in life. As Brandon's conversation with Marianne brings out, even if Brandon had overcome his gravity and reserve and had summoned the fortitude to dare to approach Marianne while her thoughts and affections were absorbed by Willoughby, he would have thwarted because of Marianne's rejection of second loves, especially second loves in "old" and "infirm" men who dare to wear flannel.
[Marianne said,] "[Colonel Brandon] may live twenty years longer. But thirty-five has nothing to do with matrimony."
Marianne's Opinion on Brandon
On the day of the group outing to the "very fine place about twelve miles from Barton," when Brandon is so suddenly called away to attend to urgent business that "cannot afford to lose ONE hour," Marianne agrees with Willoughby's diminishing pronouncement that Brandon probably invented the urgency as a rouse to avoid the "party of pleasure." Before Brandon mounted his horse to leave, he "bid [Elinor] farewell for a longer time than [he] should wish to do" but "merely bowed" to Marianne "and said nothing."
Brandon's behavior throughout this time does not bespeak the behavior of a man who is pursuing a romantic suit for Marianne's love. He doesn't see her again until they are all in London and Marianne is first desperate to hear from Willoughby and then desperate because she has heard cold and unloving things from him. It is highly unlikely that Brandon would even contemplate pursuing Marianne's affection in such an atmosphere of sensibilities. Marianne leaves London to go to Cleveland only to fall into dangerous illness--caused by her own neglect and melancholy dejection--that brings her to the brink of death. It is Brandon who, at word of Elinor's fears, volunteers to go to Barton to bring Mrs. Dashwood to Marianne's side.
Brandon Confides in Mrs. Dashwood
It is during the carriage drive back to Cleveland that Brandon pours his heart out to Mrs. Dashwood, telling her of his deep and earnest love for Marianne and sharing all the trials of his ill-fated love for Eliza. Mrs. Dashwood is deeply by her responsive sensibilities and grants Brandon her blessing in attempting to gain Marianne's love as his own, although both have realistic doubts as to Marianne's ability to respond or reciprocate. It is during Marianne's convalescence that Brandon is invited to call at Barton cottage to inquire from Marianne about her continued improvement. Even later, Mrs. Dahwood continues to encourage Brandon to make himself a guest at their home:
"I ... rather expect to see, than to hear from [Colonel Brandon] again. I earnestly pressed his coming to us, and should not be surprised to see him walk in today or tomorrow, or any day."
Mrs. Dashwood Helps Brandon
Brandon's hours at his home at Delaford were spent in repining over the disparity and "disproportion between thirty-six and seventeen." His gloomy mood upon arriving again at Barton cottage could only be lifted by Marianne's improved health and kind welcome and by Mrs. Dashwood's encouraging words.
[Brandon] had little to do but to calculate the disproportion between thirty-six and seventeen, ... [and was] in a temper of mind which needed all the improvement in Marianne's looks, all the kindness of her welcome, and all the encouragement of her mother's language, to make it cheerful. Among such friends, however, and such flattery, he did revive.
A man in this condition of sorrowful despair could not be described as a man who was engaged in continuing a gentle suit for the love of Marianne. So when does Brandon actually initiate an active suit for Marianne's love? It is doubtful that we can truly say that he ever actually does initiate an active suit for her love. It is more correct to say that proximity and Mrs. Dashwood's good efforts cause a blooming of regard in Marianne that Brandon finally plucks by proposing marriage.
Mrs. Dashwood was acting on motives of policy as well as pleasure in the frequency of her visits at Delaford; for her wish of bringing Marianne and Colonel Brandon together was hardly less earnest, ... It was now her darling object. Precious as was the company of her daughter to her, she desired nothing so much as to ... see Marianne settled at the mansion-house ... [Marianne] was born to overcome an affection ... [and] voluntarily to give her hand to another!—and THAT other, a man who had suffered no less than herself ... [who] she had considered too old to be married,—and who still sought the constitutional safeguard of a flannel waistcoat!