It's important to remember that police jurisdictions generally follow administrative boundaries (in fact it's probably always this way). A city police force will only have jurisdiction within the city limits. What you're looking for is a dataset of administrative boundaries. Generally speaking, the Office of Management and Budget, the agency responsible for delineating census boundaries, respects administrative boundaries. You won't have a census tract that includes areas both inside and outside a city limit, county line, or state border.
However, MSAs (or more accurately all Core Based Statistical Areas) are delineated using a different procedure. They look at not only the urban center, but all outlying areas that have:
a high degree of social and economic integration (as measured by commuting to work) with the urban core
This means a CBSA could include not only areas outside the city limits, but even across county and state lines, which would involve multiple police jurisdictions. It's important to remember that in many cities, particularly in the Western US, one city could have many police jurisdictions. The City of Phoenix is one part of the Phoenix Metropolitan area that includes a number of other towns contiguous with Phoenix and included in the CBSA but each have their own police force, like Glendale, Peoria, and Tempe.
For states and counties, those boundaries are easy to come by and the demographic data easy to get through American FactFinder or DataFerritt. Your best bet for cities would be to use Census Places and filter out the Census Designated Places (CDP), which are unincorporated areas with significant populations but no police force (they're likely policed by the county).
Since UCR only gives you a name for the police force, to join the data you'll need to do a text match on the name of the jurisdiction for the police force (should just be everything before "police"). Note that UCR is a voluntary program so you may have some municipalities without data.
If the demographics you care about are from the 2010 Census, you can use the Demographic Profile table, which is a shapefile you can use to map the data as well or just display the data as a table.
In general, I'd stay away from any assumptions when it comes to the applicability of CBSA or CSA boundaries to anything outside the Census. These are statistical constructs and while they try to accurate capture the extent of economic activity in urban areas, they're really just a best guess. Good luck with your project.