Study to Evaluate the Safety, Tolerability, and Pharmacokinetics of MK-1029 in Participants With Mild to Moderate Asthma

Inclusion Criteria: * If female, must be of non-childbearing potential * Have a history of mild to moderate asthma for at least 6 months * Other than asthma, in general good health * Able to perform reproducible pulmonary function testing * Is a nonsmoker and/or has not used nicotine or nicotine-containing products for at least 12 months * Have body mass index (BMI) ≥17 kg/m\^2, but ≤35 kg/m\^2 Exclusion Criteria: * Demonstrate a decrease in absolute forced expiratory volume in 1 second (FEV1) of \>20% from the Screening Visit to the Baseline Visit * Experience a decrease in AM or PM peak expiratory flow (PEF) below the Stability Limit on any 2 consecutive days prior to the Baseline Visit * Require the use of \>8 inhalations per day of short-acting beta2-agonist metered dose inhaler (MDI) or \>2 nebulized treatments per day of 2.5 mg albuterol, on any 2 consecutive days from the Screening Visit up to the Baseline Visit * Experience an exacerbation defined as a clinical deterioration of asthma, as judged by the clinical investigator, that results in emergency treatment, hospitalization due to asthma, or treatment with additional, excluded medication (other than short-acting beta agonists \[SABA\]) at any time from the Screening Visit up to the Baseline Visit * Have been hospitalized for treatment of asthma or required oral corticosteroids for treatment of asthma within the past 6 months, or has ever required ventilator support for respiratory failure secondary to asthma * Require the chronic use of high-dose inhaled corticosteroids * Have been diagnosed with chronic obstructive pulmonary disease (COPD) or any other clinically relevant lung disease, other than asthma * Have a history of any illness that might confound the results of the study or poses additional risk to the participant * Have had recent (within 4 weeks of first dose) or ongoing upper or lower respiratory tract infection * Is nursing * Have a history of significant multiple and/or severe allergies (including latex allergy), or has had an anaphylactic reaction or significant intolerability to prescription or non-prescription drugs or food