Switching between any two instruments, even two closely-related ones, is a challenging prospect. You must practice for many hours to do it well. But often people switching between clarinets (such as between B-flat clarinet and bass clarinet) are making larger changes than necessary.
The fundamental concepts in clarinet tone production are breath support, voicing, and embouchure. These should remain basically the same whether you are playing the largest or smallest members of the clarinet family.
Breath support should, in all cases, be powerful and constant. Voicing, even on low clarinets, should be high (think “cold air”). You may find the lower clarinets are somewhat more forgiving of lower voicings, and even that some pleasing effects can be achieved. But a consistently high voicing across the clarinet family pays off in intonation, evenness of tone, and ease of response.
Embouchures must adapt, but really only to accommodate different sizes of mouthpiece. In general, the larger the instrument and mouthpiece, the more mouthpiece you will take into your mouth. However, this amount can vary even between two B-flat clarinet mouthpieces. To find the correct position for each of your mouthpieces, insert a piece of paper between the mouthpiece and reed. Where the paper stops is approximately the place where your lip should contact the reed.
Beware advice suggesting that larger clarinets use a “looser” embouchure. Embouchures for all clarinets should be airtight, but not tight.
The angle of the embouchure is also important. Clarinet mouthpieces of any size are best played at a relatively steep angle (compared to, say, a saxophone or oboe), around 30 degrees from vertical. Some larger clarinets, depending on their neck curves, seem to lend themselves to a more-horizontal angle. But bringing the bottom end of the clarinet closer to you helps to achieve a more optimal position.
Fingerings are mostly the same for members of the clarinet family, but there are some exceptions and adaptions. Advancing players should consult a good fingering chart (such as Stefanie Gardner’s bass clarinet chart) for differences. (Or even better, get a private teacher.) Note in Dr. Gardner’s chart some differences from B-flat clarinet: the use of the left hand index finger vent for C-sharp6 through G6, and the special fingerings for the extra keywork for notes below E3, if available on your instrument.