Unfortunately, your example is poorly constructed to the point of sounding disrespectful. It is not the Queen who is "processed" but the cortege.
To process can be transitive but it is not common and you should avoid it because of the confusion with the meaning of To subject to or treat by a special process; to operate on mechanically or chemically:
Also the pronunciation differs:
/prəˈsɛs/ to go in procession
/ˈprəʊsɛs/ to subject to a treatment
2. transitive. To lead or carry (a person, etc.) in procession; to go along or through (a street, an area) in procession.
1959 Times 10 Dec. 14/7 The Lord of Miracles is solemnly
processed all round the city.
1968 D. M. Smith Hist. Sicily II. lii. 484 The flagellants then processed the streets as they had done in 1647 and 1773.
1998 N. Rogers Crowds, Culture, & Politics Georgian Brit. vi. 202
At Halifax..the local benefit societies processed the town ‘wearing blue silk sashes round their shoulders and cockades in their hats’.