Forwards coach Dowson will step up to the role of director of rugby as the current incumbent of that position, Chris Boyd, switches to an advisory role, based in New Zealand.
Attack coach Vesty will become head coach as he assists Dowson, while Matt Ferguson and Ian Vass continue as scrum coach and defence coach respectively.
James Craig, who made 73 appearances for Saints before retiring in 2019, will step up from the club's Academy programme to take on a first-team role that will be focused on the lineout.
The promotion of young English coaches was always the plan for Saints since they appointed Boyd in the summer of 2018.
And now that vision is set to come to fruition ahead of the start of next season.
“We completely understand Chris’ reasons for deciding to go back to New Zealand – family must always come first,” said Saints CEO Mark Darbon.
“However, we always knew this time would come and have planned and rehearsed accordingly.
"For instance, earlier this summer when Chris took an extended break in New Zealand, Phil and Sam took sole charge of pre-season.
“From the moment we first spoke to Chris about joining Northampton Saints, we worked with him to build a high-potential group of coaches.
"Over the past three years, that group have developed and grown, and we are excited for what they can achieve together.
“While we will miss having Chris around the Gardens, we are very pleased that he will maintain a relationship with the club in an advisory role, supporting the further development of our coaching group and helping us monitor the global player market.
“I would like to take this opportunity to thank both Chris and (his wife) Linda for everything they have done for Saints so far.
"It has been a pleasure working with Chris in his current role, and I have no doubt that the club will continue to benefit from his contributions moving forwards.”
Saints chairman John White added: “It’s been fantastic to work with Chris over the past four years; he transitions into his new role with the club in great shape for the future, and I am very grateful for his efforts to date.
"We look forward to what will hopefully be a successful second half of this season, before Phil and Sam step into their new roles to take the club forward.”