Overview
The answer by Nate Eldredge addresses the core question, but some clarification is in order.
There are three main kinds of reciprocal admission, two of which (the rule for new lawyers and the rule for pro hac vice admission) I've utilized myself. I've had a several colleagues who have used reciprocal admission for experienced lawyers, including one who was a graduate of a non-ABA accredited law school in California.
Also keep in mind that a few states that otherwise allow for reciprocal admission do not permit it if you were admitted based upon a diploma privilege without actually taking a bar exam in Wisconsin (the only state that has it), because a few states think you should have to go through the experience of preparing to take and then taking the bar exam at least once.
Utah and Washington State also offered diploma privilege temporarily during COVID but no longer do so. Oregon and Louisiana offered diploma privilege admission on a one time basis during COVID in lieu of what would otherwise have been a single bar exam sitting.
Many, maybe half or so, do not allow reciprocal admission if you graduated from an non-ABA accredited law school (which mostly applies to graduates of certain law schools in California).
Financial and time considerations
Keep in mind that if you are a senior lawyer in one state and are admitted in a new state, you may see your pay and the respect that your experience receives within a law firm in your new state (or when applying for jobs in the new state) take a temporary hit as employers consider the fact that you will have to take a year or two to get up to speed on the peculiarities of the new state where you are practicing law.
Even if you don't take a formal pay or seniority cut (particularly if you have some specialized area of practice that primarily involves federal law), you should expect to spend five or ten extra hours a week for the first six to twelve months that isn't billable, adjusting to the quirks of law in your new state, learning a new e-filing systems, and updating your legacy forms to the demands of your new state.
If your new firm won't pay for the extra continuing education course and state bar association membership, pay for it out of pocket and consider it the icing on the cake of your law school expenses and reciprocal admission related fees, in much the same way that you treated your bar exam preparation course fees and the time that had to be devoted to that the first time around. A year's worth of continuing education classes can easily cost $500-$1200 a year. You may need to invest in some local law legal treatises as well which cost about $100 to $300 each, if you need to get up to speed in one or more practice area that other people in your firm don't specialize in practicing, if your firm doesn't provide a budget for that. While I'm generally comfortable with online resources, state specific treatises in hard copy can very much be worth it if you have a specialized area of practice in your new state that your colleagues at your new firm don't share.
You may also want to be taking more than the minimum number of continuing legal education courses and bar association events in your first year or two, to provide feedback about your new state's law and to meet a few lawyers outside your new firm.
You may also have to practice as a de facto paralegal for a few months in your new firm in your new state, much like you did in your first state after you graduated from law school but before you were admitted to practice, while your application to be admitted to practice in the new state is processed.
Transferring to a new state where you will start practicing as a sole practitioner is not recommended as it really helps to have at least a couple of years of law firm practice in a new state to get familiar with its quirks and acquire some state specific forms. (The same is true of being a sole practitioner straight out of law school, which I did for about nine months but would not recommend as your preferred option.)
Reciprocal Admission For New Lawyers
If you have taken the Uniform Bar Exam (which includes the multi-state bar exam, a multi-state performance test, and a multi-state essay portion) and used by all but 11 U.S. states, and received a score sufficient to be admitted to practice in the new state, within the last three to five years (depending upon the jurisdiction), you do not have to retake the UBE. The passing score ranges from 260 to 270 on a 400 point scale. Most states also have a state specific law portion, often in an essay format, and basically every state requires you to take the multi-state professional responsibility exam.
UBE states:
Many, but not all, states also allow a new lawyer admitted on one state to skip the state specific essay portion of the bar exam in the new state, either entirely, or by taking a course on the non-UBE majority rules of law in the new state in lieu of taking an essay exam on the new state's state specific laws.
A few states allow a recent multistate professional responsibility exam score can be transferred as well. But usually this must be retaken in the new state.
Almost every state that allows for reciprocal admission, even if they waive both the bar exam and the MPRE, still require lawyers seeking reciprocal admission to complete the entire character and fitness review required of new lawyers in the state. But, this is usually much easier for a lawyer who took the bar exam only three to five years ago, who completed character and fitness review in the original state at the same time, than it is for a lawyer who has practices law for many years.
Typically, the new lawyer can recycle all of their original character and fitness review materials in the new jurisdiction, update it slightly to reflect anything that happened in the last few years, and tweak it slightly to reflect different character and fitness review questions and formats in the new state. But there isn't much new data to be gathered and you can mostly use the same references (now confident that your references who were already sufficient for you to pass the review the first time won't betray you the second time).
The fees, for both the application and often a separate character and fitness review and sometimes a new MPRE fee or a score transfer fee for your UBE and MPRE scores, are, however, still hefty: $250 to $1500 for the primary application fee plus the character and fitness review fees and any other fees (e.g. for a criminal record check, or the MPRE).
These are generally lower than the fees for mid-career lawyers seeking reciprocal admission, but you are typically much less affluent in the first few years after you get out of law school and often have hefty student loan payments to make as well.
Reciprocal Admission For Experienced Lawyers
Reciprocity typically involves skipping the bar exam if one has practiced for at least three to five years (sometimes within the last five to ten years) in a reciprocal state, and those years must involve the active practice of law, or service as a judge or law professor.
Many but not all states that allow a lawyer seeking reciprocal admission to skip most of the bar exam, still require the lawyer seeking reciprocal admission to retake the Multistate Professional Responsibility Exam (MPRE), a fairly short multiple choice exam that is limited to the Rules of Professional Conduct.
Almost every state that allows for reciprocal admission, even if they waive both the bar exam and the MPRE, still require lawyers seeking reciprocal admission to complete the entire character and fitness review required of new lawyers in the state. And, for a lawyer who has practiced for many years, this can be a quite cumbersome task, because it requires, for example, providing information and ideally references from every place that the lawyer has ever lived and every place that the lawyer has ever worked, a fingerprint based criminal record check, proof of payment of all traffic tickets you've ever received in your life, and disclosure of a lifetime of mental health treatment. It can be particularly complex for attorneys who have been divorced or have been parties to multiple lawsuits, or have had any professional ethics proceeds (which is a lot more likely after thirty year than three).
And, the application fee in reciprocal admission cases is often not small. It ranges from $500 to $2500, and in many case this fee is in addition to MPRE fees and a hefty fee to have the bar admission officials review the character and fitness part of your application.
Pro Hac Vice admission compared
Also, reciprocal admission as a full fledged lawyer in a new state should be distinguished from admission pro hac vice in a single case in a jurisdiction where a lawyer is not admitted to practice.
For example, suppose that your long time business client gets into a patent infringement lawsuit in both the state where you are admitted and Michigan. Typically, all the needs to be done is to affiliated with local counsel admitted in that jurisdiction who can vouch for your good character and can provided guidance on local procedural issues to serve as junior co-counsel on the case, and you can be admitted on motion for that particular case as lead co-counsel.
