The primary purpose of this study is to evaluate the pharmacokinetics (PK) and pharmacodynamics (PD) of ublituximab in participants ages 10 to less than (<)18 years and body weight greater than or equal to (≥)25 kilograms (kg) to less than or equal to (≤)40 kg with RMS (Part A) and to evaluate the non-inferiority of ublituximab compared with fingolimod in pediatric RMS participants with body weight ≥ 25 kg (Part B). The study will further evaluate long-term safety and efficacy of ublituximab in RMS in pediatric participants during its extension period (Part C).

Inclusion Criteria for Part A and Part B:

1. Diagnosis of RMS.
2. EDSS at screening: 0-5.5, inclusive.
3. Neurologic stability for ≥ 30 days prior to screening, and between screening and Week 1 Day 1 (W1D1).

Inclusion Criteria for Part C:

1. Participants must have completed Part A (Week 24 visit) or Part B (Week 96 visit) to be eligible for Part C.

Exclusion Criteria for Part A and B:

1. Known presence or suspicion of other neurologic disorders that may mimic MS.

2. Prior treatments:

   1. Systemic corticosteroids (>0.1 milligrams/kilogram/day [mg/kg/day], or >5 milligrams/day [mg/day] of prednisone equivalent) or adrenocorticotropic hormone (ACTH) within 30 days prior to the screening MRI scan (note: Topical, ophthalmic, or inhaled corticosteroids are permitted).
   2. High dose intravenous immunoglobulin (IVIG) or subcutaneous IG (SCIG) within 2 months prior to W1D1.
   3. Treatment with anti-CD20 or other B cell directed treatment at any time.
   4. Treatment with alemtuzumab, cladribine, cyclophosphamide, mitoxantrone at any time.

Additional Exclusion Criteria for Part B Only (Relevant to Fingolimod Treatment):

1. Treatment with fingolimod or other sphingosine-1 phosphate-1 (S1P1) modulators at any time.
2. The following antiarrhythmic drugs at Screening: Class Ia anti-arrhythmics.

Exclusion Criteria for Part C:

1. If the absolute lymphocyte count (ALC) is outside the specified range the participant will not be eligible to receive ublituximab in Part C.

Note: Other protocol-specified inclusion/exclusion criteria may apply