Dovitinib Lactate, Gemcitabine Hydrochloride, and Capecitabine in Treating Patients With Advanced or Metastatic Solid Tumors, Pancreatic Cancer and Biliary Cancers

Inclusion Criteria: * Part A: histologically or cytologically confirmed solid tumors that ale advanced or metastatic that the gemcitabine combination is considered standard therapy or a rational option * Part B: histologically or cytologically confirmed adenocarcinoma of the pancreas or the biliary tract (cholangiocarcinoma)that is advanced or metastatic * Part B: must have tumor lesions amenable to safe biopsy and willing to consent to tumor biopsies * Patients with at least one measurable site of disease as defined by Response Evaluation Criteria In Solid Tumors (RECIST) criteria version 1.1 that have not been previously irradiated * Eastern Cooperative Oncology Group (ECOG) Performance Status =\< 1 * Life expectancy \>= 3 months * Absolute neutrophil count (ANC) \>= 1,500 cells/mm\^3 * Platelets \>= 100,000 cells/mm\^3 * Hemoglobin \>= 9.0 g/dL * Aspartate aminotransferase (AST)/serum glutamic oxaloacetic transaminase (SGOT) and alanine aminotransferase (ALT)/serum glutamic pyruvic transaminase (SGPT) =\< 1.5 x Upper Limit of Normal (ULN) * Bilirubin =\< 1.5 x ULN * Serum creatinine =\< 1.5 x ULN * International normalized ratio (INR) =\< 1.5 (anticoagulation is allowed if target INR =\< 1.5 on a stable dose of warfarin or on a stable dose of low-molecular-weight heparin (LMW) heparin for \> 2 weeks at the first dose of study agent);if urinalysis shows proteinuria, 24 hour urine collection is to be performed and the 24 hour urine protein is to be \< 2 grams to be eligible * Willing and able to take oral medication, comply with scheduled visits, treatment plan and laboratory tests * Ability to understand and willingness to sign a written informed consent, a signed informed consent must be obtained prior to any specific procedures Exclusion Criteria: * Part B: Patients with history of another malignancy within the last three years prior to study entry, with exception of adequately treated in-situ carcinoma of the uterine cervix, or skin cancer (such as basal cell carcinoma, squamous cell carcinoma, or non-melanomatous skin cancer) * Patients who have received the last administration of an anti-cancer therapy including chemotherapy, immunotherapy, hormonal therapy and monoclonal antibodies (but excluding nitrosourea, mitomycin-C, targeted therapy and radiation) =\< 4 weeks prior to starting study drug, or who have not recovered from side effects of such therapy * Patients who have received the last administration of nitrosourea or mitomycin-C =\< 6 weeks prior to starting study drug, or who have not recovered from the side effects of such therapy * Patients who have received targeted therapy (e.g. sunitinib, sorafenib, pazopanib) =\< 2 weeks prior to starting study drug, or who have not recovered from the side effects of such therapy * Patients who have had radiotherapy =\< 4 weeks prior to starting study drug, or =\< weeks prior to starting study drug in the case of localized radiotherapy (e.g. for analgesic purpose or for lytic lesions at risk of fracture), or who have not recovered from radiotherapy toxicities * Patients who have undergone major surgery (e.g. intra-thoracic, intra-abdominal or intra-pelvic), open biopsy or significant traumatic injury =\< 4 weeks prior to starting study drug, or patients who have had minor procedures, percutaneous biopsies or placement of vascular access device =\< 1 week prior to starting study drug, or who have not recovered from side effects of such procedure or injury * History or presence of serious uncontrolled ventricular arrhythmias or presence of serious uncontrolled atrial fibrillation * Clinically significant resting bradycardia * Known left ventricular ejection fraction (LVEF) assessed by 2-D echocardiogram (ECHO) \< 50% or lower limit of normal (whichever is higher) or multiple gated acquisition scan (MUGA) \< 45% or lower limit of normal (whichever is higher) * Any of the following within 6 months prior to study entry: myocardial infarction (MI), severe/unstable angina, Coronary Artery Bypass Graft (CABG), Congestive Heart Failure (CHF), Cerebrovascular Accident (CVA), Transient Ischemic Attack (TIA), Pulmonary Embolism (PE) * Uncontrolled hypertension defined by a systolic blood pressure (SBP) of \>= 160 mm Hg and/or diastolic blood pressure (DBP) \>= 100 mm Hg, with or without anti-hypertensive medication * Previous pericarditis; clinically significant pleural effusion in the previous 12 months or current ascites requiring two or more interventions/month * Any active gastrointestinal (GI) impairment which, in the opinion of the investigator, would impair or alter the absorption of dovitinib (e.g. ulcerative colitis, or Crohn's disease) * Positive hemoccult test result within 14 days prior to the start of study treatment * Cirrhosis, chronic active hepatitis or chronic persistent hepatitis * Known diagnosis of human immunodeficiency virus (HIV) infection (HIV testing is not mandatory) * Patients who are currently receiving oral anticoagulation treatment with therapeutic doses of warfarin with goal INR \>= 1.5; patients receiving anticoagulation by subcutaneous injection such as heparin, enoxaparain, fondaparinix that are not expected to interact with study medications will be eligible * History of alcoholism, drug addiction, or any psychiatric or psychological condition which, in the opinion of the investigator, would impair study compliance * Uncontrolled diarrhea \>= Common Terminology Criteria for Adverse Events (CTCAE) grade 2 * Other concurrent severe and/or uncontrolled concomitant medical conditions (e.g. active or uncontrolled infection) that could cause unacceptable safety risks or compromise compliance with the protocol * Pregnant or breast-feeding women * Women of child-bearing potential, defined as sexually mature women who have not undergone a hysterectomy or who have not been naturally postmenopausal for at least 12 consecutive months (e.g., who has had menses any time in the preceding 12 consecutive months), must have a negative serum pregnancy test =\< 3 days prior to starting study treatment * Women of child-bearing potential, who are biologically able to conceive, not employing two forms of highly effective contraception; male not using at least at least one form of highly effective contraception will be excluded; highly effective contraception (e.g. male condom with spermicide, diaphragm with spermicide, intra-uterine device) must be used by both sexes during the study and must be continued for 8 weeks after the end of study treatment; oral, implantable, or injectable contraceptives may be affected by cytochrome P450 interactions, and are therefore not considered effective for this study * Patients with known brain metastases or who have signs/symptoms attributable to brain metastases and have not been assessed with radiologic imaging to rule out the presence of brain metastases