A space colony will be designed to be fully self-sustaining with a margin of safety once established. For example all hardware will be 3D printed from raw materials based on models from standard template library. Similarly software and administration will be based on standard templates. By the time your people reach another galaxy those template libraries should cover everything they need.
The city will import volatiles and metals for life support and production. However the locations of colonies will be selected such that the colony will be able to source these resources locally using its own integral and sustainable mining and transport capacity.
Using standard templates means that the colony will be deliberately behind the technology curve by a significant amount. This typically means that things not needed by the colony will be more efficient to do back home. Examples are research, exploration beyond local area, major system upgrades to the colony, and non-local politics. The colony would still have support capacity for all of these, but they would still come from back home with all the necessary instructions and resources attached.
Literal expansion by establishing new colonies would also be mostly managed from home as "the civilized worlds" can produce the initial colony modules and source the human resources more efficiently. The already established colony would provide logistics support in establishing the new colony. For example it might provide metals and volatiles from its own sources until the new colony establishes its own. This would probably be cheaper than shipping water from another galaxy.
People would, at least initially, feel the need to stay connected with from where they came from and "feel civilized". Many things would be imported to cover this need. Art, entertainment, luxuries, religion, politics are examples of this. None of these are really crucial enough for local production to be necessary. They all are also relatively cheap to import. A copy of an imperial proclamation or a religious speech does not really require that much extra space on a ship. And luxuries just get more valued and appreciated, if they are rare due to transport cost.
For that matter keeping the colony importing its ideas and values is something the colonial power will want. So local alternatives will be discouraged and imports subsidized. Local extensions might be welcome and even imported back home. A loop is twice as solid a link as a one way flow is. And properly nurtured it will strengthen over time so colonies will grow more connected to home instead of drifting apart.