Mrs. Walker, who is purposefully driving around the Pincio in the late afternoon during the promenade hour on intentionally to find Daisy, pursues her, in a loose sense, to lend respectability to Daisy's character. "To pursue" is not the right verb for what happens. "To seek" is a better verb choice.
Daisy Miller and Mrs. Miller visited Mrs. Walker at her residence. Near their departure, Daisy announces that she is not riding home in the Miller carriage with her mother but is rather going for a walk alone, unescorted, to meet someone at Pincio Hill, site of the ancient Roman Pincio Gardens. She confesses that the person she is to meet is Giovanelli. This alarms Mrs. Walker for several reasons: the lateness of the hour; the unescorted walk; the plan to meet an Italian man not of an acceptable social class; the plan to meet him alone and unescorted; the fact that Daisy is not walking there with her mother. Mrs. Walker calls for her carriage so she may seek out Miss Miller and persuade her to ride along with her in the carriage.
This is not the first or only time Daisy has behaved in such an ill-thought out manner. Other Americans in Mrs. Walker's social circle are beginning to talk about Daisy's casual impropriety and maids are beginning to laugh at Daisy behind her back. Daisy's mother is useless and ignorant about caution in social matters, partly because she herself was brought up undereducated and in a working class social sphere. Mrs. Walker is respectable and wishes to lend her respectability to Daisy so that she will be recognized as a respectable young lady with whom more wise and cautious people might safely associate. Mrs. Walker "pursues" Daisy, she seeks her out, to save her from her own folly and give protection to Daisy's good name.
"It seemed too pitiful, not even to attempt to save her. I ordered the carriage and put on my bonnet, and came here as quickly as possible. Thank Heaven I have found you [both]!"
Daisy is careless of the "story" appearances and her behavior tells about her good name and respectability. Winterbourne calls Daisy "innocent" while Mrs. Walker rebuts with "She's very crazy!" Winterbourne says Daisy's fault is in being "uncultivated" while Mrs. Miller declares Daisy "is naturally indelicate." Consequently when all these elements are joined together, it is clear that Mrs. Walker's practical and psychological motive for seeking Daisy was "to save her."
"What do you propose to do with us?" asked Winterbourne, smiling.
"To ask her to get in, to drive her about here for half an hour, so that the world may see she is not running absolutely wild, and then to take her safely home."