Who owns things bought with research funding after project ends?

In the US, at least, there is a generally clear distinction between two classes of things bought with funding:

• "Capital expenditure" items are individually identified in a project budget, e.g., "36-node computing cluster" or "Materials for building prototype robot".
• 'Overhead' or 'Material and Supplies' items are more routine items that are not individually identified, but are considered part of the routine cost of business, e.g., office supplies, laptops, laboratory reagents.

Capital expenditure items are typically owned by the funder, and their disposition at the end of the project is at the discretion of the funder. When this is the US government, it is called GFE - Government Furnished Equipment. In most cases, the funder lets it stay at the institution (effectively giving it to them), but not always. An institution might even be required to give the equipment to another researcher.

Overhead items are generally owned by the institution that is executing the project, and are often owned by the institution in general, rather than being associated with the particular project. These typically further break down into two sub-categories: tracked, and untracked. Tracked is things like laptops, that many institutions still consider expensive enough to keep track of who they give them to and (at least theoretically) ask for them back eventually. Untracked is things like paper and staplers, which are below the threshold where the institution cares. Again, typically all of it technically belongs to the institution, but in practice many institutions will let somebody keep a low-value 'personalized' item such as an old laptop.

The exact definitions of which things go in which categories depend on the institution, the funder, and the particular contract, but this covers most of the typical cases. Make sure you check particular regulations and customs with any particular institution before taking anything, though!

The funding agreement or regulations by the sponsor will usually clarify this. In most cases, it can be expected that the legal institution where the project is run buys these things and also owns them after the project. In some cases, the sponsor may own expensive equipment himself, so that it can go with the PI when changing affiliations.

I've also seen regulations where the sponsor only pays the depreciation of long-lasting equipment during the runtime of the project, so the institution would have to cover for the remaining costs.