Most of the literature seems to be concerned with machines with single oracles for specific problems, however there appear to be a few papers that consider machines with multiple oracles. Is there a good paper or thesis which provides an overview of what is known about such machines? In particular I am interested in P with multiple oracles.
Here is a more recent paper giving a difference between single vs. multiple oracles motivated by cryptography:
Donald Beaver and Joan Feigenbaum. Hiding instances in multioracle queries. STACS 1990. Springer Lecture Notes in Computer Science, 1990, Volume 415/1990, 37-48, DOI: 10.1007/3-540-52282-4_30
Here is an older paper you might find helpful: Logspace Machines with Multiple Oracle Tapes by Nancy Lynch at MIT (PDF). In particular Theorem 2.2, on PDF-page 5, might be the type of thing you're looking for. There's also a section on hierarchies defined by different numbers of oracle tapes per machine.
Disclaimer after the fact: Looks like a similar question and (even more similar) answer were given here.